The Scotland international looked set to join West Ham earlier in the January transfer window, but the move broke down on Saturday.
That allowed the Baggies and Tony Pulis to step in, and they secured the deal late on deadline day.
The 31-year-old made 342 appearances for the Red Devils during his time at Old Trafford, and was originally named as Louis van Gaal's vice-captain at the start of this season.
But Michael Carrick's return from injury saw him dropped to "third captain", and he has made just four Premier League starts this season.
He becomes Pulis' second signing at the Hawthorns after Callum McManamana joined form Wigan Athletic.
