Ole Gunnar Solskjaer's comeback kings fell short on the road and, having been top just 19 days ago, second-place United now trails Manchester City by seven points. City also has a game in hand.
Mbaye Diagne got West Brom off to a dream start after just 80 seconds before United drew level through Bruno Fernandes' stunning effort.
Harry Maguire had a last-gasp header superbly turned on to a post by Baggies goalkeeper Sam Johnstone.
Having won all of its previous seven away league matches in which United conceded first, Solskjaer will be frustrated by his side's poor response in this costly draw at the Hawthorns.
Diagne scored his first goal for the club, his hand in Victor Lindelof's face overlooked, but Fernandes superbly leveled at the end of the first half. He fired home a stunning leveler in the 44th minute, meeting Luke Shaw's cross with a fine hooked effort that flew into the top left-hand corner.
The second half was more open and eventful.
Referee Craig Pawson overturned his decision to award a penalty for Semi Ajayi's foul on Maguire after reviewing footage on the pitchside monitor, but offside would have come into play anyway if he had stuck with his decision.
Johnstone denied Mason Greenwood before Darnell Furlong cleared Scott McTominay's effort off the line, while at the other end David De Gea prevented Diagne securing a shock win with a superb double save.
Diagne only had himself to blame in the 78th minute as he somehow failed to score. Having shaken off Maguire — perhaps illegally — to go through one-on-one with De Gea, the ball looped up after his initial shot was saved and Diagne looked set to nod into an empty net until the goalkeeper got back superbly to claw away.
