Goals from Phil Jones, Wayne Rooney and Danny Welbeck gave United back-to-back league wins for the first time since December, following their 2-0 success at Crystal Palace last month.
Victory took the defending champions to within nine points of fourth-place Manchester City, although they have played two more games, while Pepe Mel’s West Brom remain a point above the relegation zone in 17th place.
United’s last game at The Hawthorns had been Alex Ferguson’s last as manager and his successor, David Moyes, was looking for a reaction to his side’s shock 2-0 loss to Olympiakos in the Champions League.
After Marouane Fellaini had seen a header cleared off the line and West Brom goalkeeper Ben Foster had tipped a header from Rafael da Silva onto his crossbar, United broke through in the 34th minute.
A free-kick wide on the right gave Robin van Persie an opportunity to whip a cross into the box and Jones stole in front of a clutch of defenders to beat Foster with a glancing near-post header.
In reply, Claudio Yacob and Victor Anichebe each went close for West Brom, but Foster was fortunate to escape censure after he miscalculated an attempt to control a lofted pass from Juan Mata outside his area and the ball struck him on the hand.
West Brom forced United onto the back foot early in the second half, with Zoltan Gera testing David De Gea at his near post and Mata hooking away a low cross from Anichebe.
However, United tightened their grip on proceedings in the 65th minute, with Rooney nodding in a cross from Rafael at the back post to claim his 11th league goal of the campaign.
Gera tested De Gea again as West Brom probed for a path back into the game, but Welbeck killed off the home fans’ hopes of a comeback in the 82nd minute when he ran onto Rooney’s through ball to curl a shot past Foster.
Later on Saturday, Chelsea will look to extend their lead at the top of the table by beating Tottenham Hotspur at Stamford Bridge.
A win would lift Chelsea seven points above Liverpool and Arsenal, who both have a game in hand, and nine points clear of fourth-place Manchester City, who will have three games in hand.
Elsewhere, bottom club Fulham visit second-bottom Cardiff City, while Crystal Palace host Southampton and Stoke City visit Norwich City.