Marco Silva’s side looked on course for only their second last-four appearance since they lost to West Ham in the 1975 final when their world collapsed around a handball incident 15 minutes from time.
Jadon Sancho seemed to be shooting an equaliser into an empty net, only for Willian to charge across and clear.
Referee Chris Kavanagh initially gave a corner, but VAR advised the official to check the pitchside monitor, with Willian protesting and manager Silva furious.
Silva was red carded before Kavanagh could even reach a decision. Once he did, Willian was dismissed as well. Then Aleksander Mitrovic, who had put Fulham in front, pushed the referee’s arm as he angrily remonstrated and became the third visitor to be sent off.
Once order was restored, Bruno Fernandes tucked home the penalty.
In the space of two minutes, Fulham had lost two men, their manager and the lead.
United will now play Brighton in the semi-final at Wembley.
The sight of Silva pacing up and down the tunnel in a state of fury did not suggest it is a defeat he will take well.
Fulham impress – until mayhem breaks out
The carnage of the handball incident completely changed the direction of a game that Fulham had dominated.
Not since the days of Alan Mullery and Bobby Moore have they enjoyed the experience of an FA Cup final, and they appeared so close.
Had it not been for David de Gea, they would have been out of sight.
After Mitrovic had pounced at the far post to convert Issa Diop’s flick on, the United keeper performed heroics to keep the score down.
At full stretch, De Gea turned away a curling Willian shot that was heading for the far corner, then denied Mitrovic with another superb stop before the Serbian headed a half-decent opportunity over.
With former United midfielder Andreas Pereira excelling on his Old Trafford return and Tim Ream leading a robust defence, the hosts appeared to be running out of ideas.
That was until Antony led a counter-attack and squared to Sancho, who skipped past Bernd Leno, turned inside Harrison Read and steadied himself before shooting – with Willian getting in the way on the goalline.
After that, it was mayhem.
United making it hard for themselves
We can never know what would have happened if Fulham had not suffered such a disastrous couple of minutes.
It is beyond question, however, that up to that point, United were poor.
Maybe their increasing number of below-par performances is a duel consequence of an arduous post-World Cup fixture schedule in which they have played 13 successive midweek matches (with a minimum of four more to come after the international break) plus a growing number of injuries and suspensions.
The continued injury-enforced absences of Christian Eriksen and Anthony Martial has put increased pressure on others to perform.
One who has risen to the challenge is Casemiro, but on Sunday the Brazilian was serving the first game of a four-match ban for his second dismissal in a matter of weeks.
It is to their credit, therefore, that they keep managing to find a way to win quite so often.
Fernandes, in particular, drives United on from central positions and Shaw has returned to form after a difficult few games.
However, without De Gea, they may have been out.