United lost Spain international Juan Mata to two yellow cards inside two minutes midway through the first half and went behind to a sharp goal from Venezuelan striker Salomon Rondon on 67 minutes.
Louis van Gaal's side were unable to fight back and slumped to their first league defeat at The Hawthorns since 1984 to end their recent run of four straight wins in all competitions.
"When you have to play 75 minutes with 10 players, I don't think we were so uncomfortable," United manager Van Gaal told BBC Sport.
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"The loss is a big blow because it was not necessary. When you see the first half it was not necessary, I think."
United remain sixth, three points adrift of Manchester City in fourth place, with Liverpool breathing down their neck in seventh after fighting back from 1-0 down with 10 men to consign Palace to a 12th league game without a win.
The home side went ahead through Joe Ledley's low drive just after the break and Liverpool found themselves further on the back foot when they lost James Milner to a second yellow card on 62 minutes.
Yet Juergen Klopp's men kept pressing and their perseverance paid off when Palace goalkeeper Alex McCarthy lost his footing and scuffed a clearance straight to Roberto Firmino, who fired it past him for the equaliser on 72 minutes.
Deep into stoppage time late substitute Christian Benteke surged into the left side of the Palace penalty area and went down after a slight contact with defender Damien Delaney, who was sliding on his knees.
"ROBBED"
Referee Andre Marriner awarded the spot kick after consulting his linesman and Belgium's Benteke opted for a stuttering run-up before burying the penalty with virtually the last kick of the match.
"I feel sorry for Palace but I have just seen the replay in the dressing room and it was a clear penalty," Klopp told BBC Sport.
"If there was no contact, Benteke would have been in a great position to shoot. He made the move and forced the situation.
"This is the third league win in a row. I can't believe how it feels -- really good, a big moment for us."
Former Aston Villa striker Benteke was adamant he had been floored, saying: "I believe he touched me, otherwise I don't go down."
But Palace manager Alan Pardew threw his jacket into the dugout in disgust at the penalty decision and said afterwards his side were "robbed".
"You've got to be 100 percent sure," he told Sky Sports. "In super slow motion the knee touches the foot and he goes over like he's been shot.
"The linesman assumes that little touch is a penalty. If you think that touch affects the fall and the way he lands, then you really need to be consulted."
Palace, who are without a league win in 2016, are 15th in the table, nine points above the relegation zone.
Pardew told BBC Sport: "I feel like we've been robbed a little bit but again maybe we shot ourselves in the foot a bit."
(Editing by Toby Davis)
