Allan has put pen to paper on a deal, with a further 12-month option, which will keep him at The Hawthorns until at least the summer of 2014.
West Brom agreed a fee - believed to be £300,000 - last month with United, and the player has opted to move to Albion after holding talks with head coach Roy Hodgson and passing a medical.
Scotland Under-21 international Allan was also linked with a possible move to Old Firm rivals Celtic and Rangers but has opted to pursue his career south of the border.
Hodgson, who met with Allan, 20, and his family on Friday, is confident the player can have a big part to play in Albion's long-term future.
He said: "We've followed Scott for a long time, we've pursued him, and we've made every effort to persuade him to come to the club and are delighted it has happened.
"He will be a very good addition to our squad. This is not just a player we're bringing to help for the next four or five months.
"This is a player that we think, in the next few years, will be an important player for the club."