Young striker Anthony Elanga ballooned his shot over the bar to give Chris Wilder's side a win that was celebrated in enthusiastic fashion by a 9,000-strong visiting support.
However, they only had themselves to blame after failing to take numerous chances in a game where they had 30 shots at goal, the clearest of which came when Cristiano Ronaldo missed a penalty in the first half.
Jadon Sancho seemed to have put United on the road to the last- 6 when he marked his FA Cup debut with a goal.
Instead, United's most realistic chance of silverware has now disappeared.
United manger Ralf Rangnick later said: "Incredibly disappointed, we should have won and killed off the game in the first half. Crossbar, post, missed penalty and numerous chances. It should have been 2-0 or 3-0. We conceded a goal which I don't understand why it stood. He controlled it with his hand.
"In the moment when they scored, it was clear that VAR would not allow this goal to stand. We were not well positioned in that moment, too open and allowed them the counter, we should have won the game.
"Anthony Elanga, anyone can imagine how he us, he is shattered and absolutely disappointed. It was the eighth penalty which had all converted."