A controversial penalty converted by Jarrod Bowen on 92 minutes earned the Hammers the victory.
Before then, Crysencio Summerville had put the Irons ahead before Casemiro headed home an equaliser.
But West Ham were handed a penalty by VAR after an apparent foul by Matthijs de Ligt on Danny Ings.
United manager Erik ten Hag said later: "Three times this season we feel injustice. We have to score, we created so many chances. We should've been two or three up. Second half we were forcing it but we allowed them into the game. When you are losing 1-0 you need big personality and character of the team and showed resilience to get back into it. Unfair and unjust the way we conceded the penalty.
"It was so difficult to see. Before the season there was the instruction about VAR only interfering in clear and obvious mistakes. That is definitely not a clear and obvious mistake from the on field referee.
"More frustrations but I can do nothing with that. They don't collect points and that's what we have to do. We have to look in the mirror, we don't score in a good game from our side. Create loads of chances and concede none but when you lose in this way it's a bad feeling.
"I spoke with them (the match officials). But the decision is made. There's no way back and that's football. That's a third time I have felt injustice in the season and it has a big impact on our team and on our scores and where we are in the table. It's not right."