The decision comes just 14 months after the console's launch, during which time BBH's advertising has won a spate of awards.
It gives McCann-Erickson a clean sweep of Microsoft's advertising business; the agency is the long-time incumbent on the company's main software account.
McCann is expected to formally take control of the Xbox work in the summer.
Its longer-term challenge will be to promote Xbox Live, the console's broadband gaming service.
As the decision was being confirmed last week, the brand's 'Champagne' spot won the Advertising Club of New York's Andy award, which recognises creative excellence from advertising across the world.
However, the execution, which had already pocketed the Cannes Gold award, was banned in the UK by the Independent Television Commission last June after the organisation upheld 136 complaints that the ad was offensive and shocking.
The ad showed a woman giving birth to a baby boy and follows a surreal sequence in which he shoots out of a window, ages rapidly as he travels at speed through the air before violently crashing into his own grave as an old man. The ITC said at the time it did not agree with the advertiser's claim that it conveyed a "positive statement" about life.
Microsoft has attempted to position Xbox as a more inclusive gaming brand under the tagline 'Play More', contrasting with what it characterised as Sony's darker, more exclusive positioning of its PlayStation 2 console.
Microsoft has spent £3.3m on advertising the brand in the UK in the past 21 months.
While BBH's advertising has attracted critical acclaim and controversy in equal measure, Xbox's first-year sales have failed to keep pace with Sony. According to Screen Digest, it shifted 550,000 consoles in 2002, compared with PlayStation 2's 1.93 million.
Xbox changed its strategy from promoting its hardware platform as a brand to pushing specific games titles in the run-up to its first Christmas season last year.
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