Christian Benteke’s stoppage-time penalty gave Liverpool a come-from-behind victory.
The Belgian converted after being taken down by Damien Delaney in the closing seconds at Selhurst Park.
Joe Ledley gave Palace the lead only to have Roberto Firmino equalise after James Milner was sent off for a second yellow card.
Liverpool’s third straight Premier League win moves them up two spots to seventh.
“It was an incredible performance from all of us, especially after the red card,” Liverpool defender Dejan Lovren told Sky Sports.
“We stepped up and showed our skills and character.
“This is a lift for every one of us. It’s good for confidence and we need to continue like this.”
Palace are now winless in 12 in the league and remain in 15th place, not yet clear of relegation.
“I feel like we’ve been robbed a little bit but again maybe we shot ourselves in the foot a bit,” Palace manager Alan Pardew said.
“When they went down to 10 men maybe we should have pushed on a little bit.”
While Reds manager Juergen Klopp said he felt the penalty call was correct after watching a replay, Pardew disagreed.
“Damien Delaney literally pulls out of it and the follow through of his knee gets it,” he said.
“If you give a penalty for that, blimey that’s a bit tough.”
Salomon Rondon scored the only goal to give West Brom their first home league win over Manchester United since 1984.
“It’s a record we knew about before the game and were desperate to put it right,” West Brom’s former United midfielder Darren Fletcher said.
“I thought to a man today our team performance was fantastic all around the pitch.”
Manchester United’s Juan Mata was sent off after picking up two yellow cards in the first 26 minutes.
The visitors stayed on even terms until Rondon finished off a cross from Sebastien Pocognoli midway through the second half.
The loss kept United from going level on points with fourth-placed Manchester City, who hold the final Champions League spot.
“It’s a setback,” midfielder Michael Carrick said. “We wanted to carry on the form and results we have shown of late so it’s very disappointing… losing a man like we did makes it even more difficult.”
The win lifts West Brom up to 11th and within a point of manager Tony Pulis’s goal of 40 to ensure Premier League status for next season.
“The most important thing is the next game,” Pulis said. “As you’ve seen this year, results have been off the radar.
“Let’s get 40 points, I think that’s definitely safe.”
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