In the war for world technology domination, Google is quietly establishing a beachhead in Oakland. This spring delivery trucks have been pulling up to the city’s public schools and dropping off dozens of laptops — a total of about 10,000 across the city — that use Google’s Chrome operating system rather than those made by Microsoft or Apple. Sleek but spartan, and highly economical at $230 each, the Chromebooks run web applications — and Google just happens to have a suite of options designed for the operating system — on their browsers rather than using traditional software. It’s paving the way for a dramatic change: a new generation of kids who do their homework in Google Docs and may never have even heard of Microsoft Word.

What in the name of Bill Gates is going on? Sure, Google’s incredible ascent in mobile devices — its Android is now the No. 1 operating system globally for tablets and smartphones — has been well chronicled. But even as that has been happening, the company’s Chrome OS has begun chipping away at the big players.

Then there’s Chrome’s sibling browser — now the most used web navigator on the planet. Derided as a long shot when it launched in 2008, the Chrome browser boasts a speed and simplicity that have attracted hundreds of millions: Today it has nearly twice as many users as Microsoft’s once seemingly unbeatable Internet Explorer (IE), whose market share has shriveled from about 68% to 25%, according to StatCounter. (Three other trackers report similar numbers. A fourth, Net Applications, which counts unique users rather than total usage, still ranks IE as the most-used browser.) The Chrome browser helped remake PCs into devices on which web apps such as Google’s Gmail, Docs, and Maps could run faster and more easily, making them viable alternatives to desktop software from Microsoft and others.

Google has pushed its web-centric vision further with the Chrome operating system. As with the browser, critics scoffed when the OS made its debut — “no chance,” one tech critic predicted; “doomed,” proclaimed another — and the first Chromebooks were clunky. No longer. John Krull, the information officer for the Oakland Unified School District, says Chromebooks are far easier than alternatives when it comes to things like maintenance and configuring them for state-mandated tests. Since they run only web-based programs, they require no software downloads or updates, and are so easy to troubleshoot they often require no additional IT staff.

Chromebooks have been surging in popularity. Over the holidays in 2013, two Chromebook models were the No. 1 and No. 3 bestselling laptops on, and they’re being adopted in schools and businesses around the world. The NPD Group estimates that in February, they accounted for 5.4% of notebooks sold at U.S. retail outlets, up from 3.5% a year earlier. In sales through resellers to businesses and educational and government institutions, Chromebooks accounted for 33.5% of notebooks, up from 3.6% the year before. Says Stephen Baker, vice president of industry analysis at NPD: “They came out of nowhere in the last 15 months and are selling like crazy.”

Chromebooks are made by the likes of Dell, Hewlett-Packard, Acer, Samsung, and Lenovo, and Google earns no direct revenue from the laptops, the operating system, or its free browser. But Chrome and Chromebook users on average run more searches and spend more time using Google’s ad-supported services, and are more likely to buy music and movies from Google’s Play store. “Chrome is extremely important and strategic,” says prominent venture capitalist Ben Horowitz, who once was an executive at browser pioneer Netscape.

Now Google is extending Chrome technology into new areas, including mobile devices, television, and the Internet of things. After releasing a Chrome browser for iPhones and Android devices, Google introduced Chromecast, a gizmo that resembles a thumb drive (it’s called a “dongle”), which attaches to a television set and allows it to play, or “cast,” anything that’s happening on your desktop or mobile browser. With millions sold, the $35 device has given Google a firm toehold in the living room, where it is battling other providers of streaming-media devices like Apple, Amazon (which just announced its Fire TV player), and Roku. In February, Google unveiled Chromebox for Meetings, a $999 videoconferencing system for office use that includes a camera, speaker, and remote control.

Pundits have long argued that Chrome and Android will collide, creating overlap and inefficiency, as mobile and nonmobile devices interact more and more. But Sundar Pichai, the Google executive in charge of both Chrome and Android, dismisses the objection. He says the two are already working in harmony. (Chromecast is a good example: An Android-based phone can control the Chrome-based device.) He predicts that the walls between Google’s two operating systems will disappear. Android gets most of the attention as the linchpin of Google’s mobile push, but Pichai says Chrome will be vital as computing extends to thermostats, fire alarms, cars, wearable devices, and more. As he puts it, “Chrome will play a pivotal role in a multiscreen world.”

Google built Chrome to bend the arc of tech history in its favor. A cloud-computing pioneer, Google bet early that virtually everything we do on PCs would eventually be delivered over the Internet. A decade ago it began developing web apps to rival Microsoft’s software cash cows like Word, Excel, PowerPoint, and Outlook. But Google believed the transition to the cloud wasn’t happening fast enough, in part because web browsers like Internet Explorer, Mozilla’s Firefox, and Apple’s Safari were slow to innovate. Chrome was its answer: It leapfrogged rivals in speed, security, and capabilities.

The success has caused rivals to ramp up their capabilities, which Google says was its hope all along. The company made Chrome “open source” so rivals could see and appropriate its innovations. Says Linus Upson, who leads the Chrome engineering team: “All of our products and services get better if you are using a better browser.”

Chrome had another, less public mission: to defend the search engine that accounts for most of Google’s $59.8 billion in 2013 revenues and $12.9 billion in profits. Since 2000, Google had distributed a browser toolbar that made its search engine the default on IE and Firefox. (The toolbar also allowed Google to track users’ surfing habits.)

The toolbar — as geeky and benign as it sounds — was a key locus of Google’s combat with Microsoft. Pichai was the group’s leader, and by the mid-aughts he worried that Microsoft would modify IE to make it more difficult, or even impossible, for users to install the toolbar. In a series of sometimes tense conversations with top brass around the time of a major update to IE in October 2006, Pichai argued that Microsoft could threaten a sizable chunk of Google’s business, according to two executives who were there. “It was a doomsday-like scenario,” one of the executives says. Shortly after, Google execs gave Chrome the green light.

Because Pichai, 41, had impressed his bosses with his toolbar strategy, which helped Google cement its search dominance, he was a natural pick to lead the Chrome team. Pichai manages to be both soft-spoken and well liked but also demanding. A native of the southern Indian state of Tamil Nadu, he attended university in India before obtaining a graduate degree in engineering and materials science at Stanford. He earned an MBA at the Wharton School and worked at McKinsey and Applied Materials before joining Google in 2004.

Pichai proved to be the right choice. In Chrome he delivered a web browser that impressed even Google’s exacting founders, Larry Page and Sergey Brin. His star rose rapidly. In addition to Chrome, he took charge of Google’s apps, and after Page became CEO in 2011, Pichai became one of his most trusted lieutenants. Last year, when Android creator Andy Rubin turned his attention to a new robotics venture within Google, Page added Android to Pichai’s portfolio.

As the head of Google’s two computing platforms, Pichai is the point person in the company’s rivalry with both Microsoft and Apple, and arguably no one other than Page himself has more influence over the direction of Google’s technology. Pichai has become prominent enough that his name was publicly bandied about as a possible successor to Microsoft CEO Steve Ballmer, who has since stepped down.

Microsoft offered a tribute, albeit unintentional, to Chrome last year: It released a series of ads disparaging Chromebooks. In the ad campaign, orchestrated by Mark Penn, the Microsoft executive who once was Hillary Clinton’s strategist and pollster, actors dismiss the Google-powered laptops for not being able to run Windows or programs like Office. “Don’t get Scroogled,” they warned, claiming that Chromebooks are “pretty much a brick” when not connected to the Internet. The campaign was widely panned and considered ineffective, and has since been discontinued.

Microsoft would not make anyone available to discuss Chrome. But its public relations agency, Waggener Edstrom, provided a lengthy series of talking points that denigrated Chromebooks and their ilk as “more like a tablet with a keyboard” or the “21st-century version of a dumb terminal.” Microsoft also noted that consumers can now buy Windows-based PCs at prices comparable to those of Chromebooks and get more for their money. It laid out what it claims is a series of additional shortcomings of Chromebooks: They’re not good for some classes of apps like games; Google offers no guarantees of support; and some peripherals, such as fitness devices, may not be compatible with Chromebooks.

Much of what Microsoft ridicules as shortcomings fans see as assets. Chromebooks run web-based software that doesn’t have to be installed or upgraded, and the use of web software means data is always backed up. The notebooks boot up quickly and can be shared easily among different people, making them ideal for classrooms.

To the extent that some of Microsoft’s criticisms were valid, Google has been working to address them — for example by making Chrome OS capable of running apps offline. A growing number of developers, from publications like The Economist to software giants like Autodesk, have developed Chromebook-specific apps. There’s an added benefit: They can also run on any computer equipped with the Chrome browser.

Chromebooks remain a relatively small part of the PC market, but that could change. “They could be the vehicle to make a major dent in Microsoft’s dominant position on the desktop,” says David Yoffie, a Harvard Business School professor who has studied browsers and web competition for decades.

Despite Chrome’s success, does any company really need two operating systems? Pichai shrugs off the concern: “It’s a question that gets asked externally more than internally.” He points to Chromecast, which grew out of Google TV, an Android-based streaming-video system introduced in 2010. Google TV never really took off, but engineers on the team had an important realization. Online video apps and channels are really easy to use on computers and mobile devices. So why try to cram the smarts of an Internet TV into a new box or television set, only to have to operate it through a clunky remote control?

Google answered the question with Chromecast. The device can mirror on the TV set what’s happening on the browser on your phone, tablet, or computer. For apps that have been adapted for it, like YouTube and Netflix, it can pull the signal directly from the Internet while still being controlled from whatever device you are using. Netflix, whose service runs on more than 1,000 different smart TVs and TV boxes, says Chromecast stands out for its ingenuity. “It was bold and innovative,” says Greg Peters, chief streaming and partnerships officer at Netflix.

Chromecast demonstrates the versatility of Chrome’s technology and its role as a complement to Android, says Pichai: “The concept that you use your phone as the primary way to control your television is very, very powerful.” Google has already used similar technology to connect printers to mobile devices, and plans to use the concept of “cast” to connect mobile devices and cars to do things like play music and display navigation systems. Beyond that, Pichai is coy. “There are lots of possibilities for where we can take Chromecast,” he says. But given his track record, it’s likely that millions more people will soon be using Chrome technology.

This story is from the April 28, 2014 issue of Fortune.