Last May, Burberry CEO Angela Ahrendts flew to California from her London headquarters to introduce herself to an executive she thought could be critical to the future of her business: Salesforce.com CEO Marc Benioff. When the two met at the Ritz-Carlton in Half Moon Bay, they stood in the hall batting around ideas for 15 minutes before even sitting down. Ahrendts explained her vision: to create a company where anyone who wanted to touch the brand could have access to it. She just needed a digital platform to make it happen.

Benioff sketched a diagram of how Burberry could become a “social enterprise,” overlaying technology like Salesforce (CRM), SAP (SAP), Twitter, and Facebook (FB) atop the entire company. (Benioff signed the drawing “Angela + Marc = LIKE,” and Ahrendts keeps the framed original, pictured below right, in her office.) “I told him, ‘I think I finally met someone who talks faster and has more energy than I do,’ ” she says. “We just connected.”

The two might seem an unlikely pair—the brash tech entrepreneur and the polished CEO of a 156-year-old luxury retailer—but Ahrendts has been boldly reinventing Burberry’s image and operations since she arrived at the British company six years ago. Her moves have paid off handsomely: Annual sales of some $3 billion are more than double 2007 levels, and the stock has returned nearly 300% since Ahrendts’ arrival, while Britain’s FTSE All-Share Index is up 24%.

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To keep Burberry growing, Ahrendts recognized she needed to woo younger and more global consumers, who communicate and share information — and shop — in the digital world. She set out to develop a comprehensive technology strategy, one that utilizes obvious tools such as Facebook and Twitter, and also incorporates enterprise software from companies such as Salesforce and SAP.

The approach makes Burberry a standout in the luxury business, which has historically shied away from technology for fear of eroding its aura of exclusivity. “What they’ve done, that no other organization in the fashion industry has done, is put a relentless focus on digital innovation,” says Maureen Mullen of tech think tank L2.

The employees scuttling around Burberry’s Horseferry House in London are too well dressed to allow the company’s headquarters to be mistaken for a tech startup, but the building does exude the same kind of feel. Seventy percent of the employees at Horseferry are under 30 and are encouraged to peruse Facebook and Twitter during work hours. Ahrendts, 51, spearheaded the office’s free-lunch perk à la Silicon Valley.

Ahrendts likes to say that the members of her young employee base act as her interpreters for today’s digital world. But Ahrendts herself is an early adopter, often discovering the latest tech advancements through her three children. She was one of the first in the office to acquire an Apple (AAPL) iPad, and she has since outfitted staff across corporate offices and all mainline stores with the device. (Not surprisingly, Burberry makes a line of iPad cases.) During a recent visit by Fortune to Burberry’s offices, Ahrendts Skyped with a group of Burberry scholarship winners at Indiana’s Ball State University, her alma mater. She told the students to check out an article about Zynga (ZNGA) CEO Mark Pincus. If they had trouble tracking it down, they could find the link on her Twitter feed.

Technology plays heavily into employee communications too. Ahrendts and chief creative officer Christopher Bailey do regular webcasts for her workforce, and recently decided to up the frequency from quarterly to monthly. Her talk at a leadership conference in Atlanta in May was streamed at Burberry’s offices. Reg Sindall, executive vice president of corporate resources, jokes that at Burberry “we film ourselves filming ourselves filming ourselves.”

Burberry’s real tech cred (it has attracted employees from Nokia (NOK) and Microsoft’s (MSFT) Xbox division) comes from its organically digital approach to virtually everything it now does, from fashion shows to employee communications. “It should never be on a list: ‘Have we remembered to do the digital thing?’” says Bailey.

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The company uses Salesforce’s Chatter platform — think of it as a corporate version of Facebook — a tool that Ahrendts says has accelerated communication throughout the entire operation. Sales teams in the stores, for example, were among the first to notice that larger male customers were unhappy with the fit of one style of suit. Headquarters heard about it via “Burberry Chat,” and the design team made some alterations.

The company is also beefing up its corporate software systems, which, Ahrendts says, will ultimately result in better customer experiences. A global integration of SAP technology is almost complete, and at SAP’s Sapphire Now conference in mid- May, Ahrendts and SAP co-CEO Bill McDermott announced a new, jointly developed app that will put all of Burberry’s product and customer information into sales associates’ hands, allowing them to look up a customer’s name to see a profile that includes global transaction history and social media activity as it relates to Burberry.

Enterprise software may not be glamorous, but the luxury retailer is committed to investing in such seemingly mundane “back office” systems. Ahrendts says she’s opened Burberry up to its tech partners as an R&D lab of sorts, in an effort to make the operations side of the business as admired as the front end. “Because this is luxury and where people have grown up, in the past people have thought that perhaps creative needs to lead the way, and therefore what investment there is needs to go there, rather than into the back end,” says EVP and CFO Stacey Cartwright. “We’re very balanced.”

When Ahrendts arrived from Liz Claiborne in 2006, Burberry, which had licensed its name around the globe, lacked a cohesive image, and was underperforming the luxury market. (Retail and wholesale sales were up just 2.2%, vs. the sector’s roughly 13% the year she arrived.) Ahrendts started buying back the licenses and moved to find the right positioning for the brand.

Ahrendts and Bailey recognized that the company’s British heritage and its iconic outerwear (Ahrendts likes to say the company was born from a coat) had to play a key role in defining the company going forward. But they also decided to pursue millennials, a group its peers were ignoring. It was an emerging-markets play as well, with the company’s research showing that high-net-worth individuals in the developing world were 15 years younger than in markets such as the U.S. and Britain.

But how to communicate with this new demographic? “What is their language?” asks Ahrendts. “And that’s when we looked at each other and said, ‘It’s digital.’ ” Burberry started filtering everything through that lens. “We just naturally started asking ourselves on every single thing we did, how do we make it more connected, how do we make it more digital?” Ahrendts explains.

She points to her own kids as an example. A few weeks ago when her niece was in town visiting, Ahrendts told her teenage daughter to have her cousin meet them for brunch in an hour. Rather than pick up the phone, she chatted with her on Facebook. “She knows she’s on Facebook because they never turn it off,” says Ahrendts. “Doesn’t call, doesn’t e-mail. That’s their English. That’s how they communicate.”

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Today, Burberry has more Facebook and Twitter followers than any other luxury brand of those tracked by consultancy Stylophane. The company spends about half its media budget on digital but concedes it isn’t easy to calculate a return on the resources poured into Burberry’s technology.

To be sure, many of Burberry’s social media efforts aren’t designed to spur immediate sales. Take its artofthetrench.com platform, a compilation of photos that people have sent in of themselves wearing trench coats. “We always do something innovative with our most iconic product,” explains Ahrendts. Likewise, Burberry’s website, dubbed Burberry World, is more than an e-commerce site or even just a digital representation of its stores, which is how many retailers think of their online strategy.

Burberry is pushing to make its website a multimedia experience. Today it features an “acoustic” section with music videos from undiscovered British artists (dressed in Burberry clothing, of course). Consumers can also view beautifully edited clips of events, such as a recent production tied to a store opening in Taipei, staged and choreographed with the online video audience in mind.

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For the Taipei event, which was one part party, one part fashion show, Bailey had built a massive, 360-degree multimedia display that showed a movie of models in Burberry clothing. The online video stitched together footage of the arrival of local celebrities, scenes of guests snapping photographs of the movie on their smartphones, and other clips, all against a stirring soundtrack (“Numb” by Marina and the Diamonds).

The video had more than 130,000 views on YouTube at last count — impressive considering it’s essentially an ad that people choose to watch. In a meeting back at headquarters Ahrendts wonders if Burberry should play the video inside the Taipei stores. In other words, how could the company bring this digital representation of a physical event back into the physical world?

The company increasingly is pushing its platforms beyond branding. Burberry’s Runway to Reality lets viewers watch a live stream of its shows, order what they just saw, and receive their purchase in six to eight weeks.

Customers aren’t the only ones experiencing changes as a result of Burberry’s tech push. While Bailey was preparing the show, the company flew members of its customer service staff to London to train them on the new products. “The business model suddenly starts to change because we now have to train our sales associates on a collection that doesn’t physically exist yet,” he explains.

There’s a risk, of course, to creating a digital presence that isn’t completely transaction-oriented. “You have to walk away from the online experience thinking, ‘I need to buy that shirt’ — not ‘That’s a really cool video, but I don’t remember the product,’ ” says Duke Greenhill of luxury marketing and branding consultancy Greenhill+Partners.

For all her enthusiasm for digital, Ahrendts never loses sight of the very “analog” products that make Burberry what it is: clothing and accessories. The room practically crackled with excitement as she inspected the short trench coats that the men’s wear team had in the works. “Whatever you call it,” she said, “you want it to be as memorable as the trench. Just own it.” With that she jetted off to her next meeting. After all, Ahrendts has proved that, just like her tech partners, she knows how to move fast.