The deal was announced late on Monday, Fletcher signing a two-and-a-half-year contract with a 12-month option at The Hawthorns.
Fletcher's move to the Midlands ends his 20-year association with United.
The 31-year-old made 342 appearances for the Premier League giants but struggled for game time under manager Louis van Gaal, starting just five matches since the Dutchman replaced David Moyes at the start of the season.
During his time at Old Trafford, Fletcher won five Premier League titles, the UEFA Champions League and four FA Community Shields among others.
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