Elsewhere, Southampton beat Brighton, West Ham won at Watford and Sheffield United lost at home to Leicester.
In the Premier League’s late kick off, Liverpool dominated against Arsenal, winning 3-1.
A Mohamed Salah brace guided the Reds past the Gunners at Anfield.
Palace’s shock win at United was the result of the day and Ole Gunnar Solskjaer was furious with his players after the game.
Match of the Day running order: When Man Utd, Liverpool, Arsenal and Chelsea will be on (Image: GETTY)
Match of the Day running order (Image: GETTY)
Solskjaer, who watched his side miss another penalty after Marcus Rashford failed to score from the spot, said: “They scored 2 and we scored only one.
“We just were not clinical enough in front of goal and they scored two very easy goals.
“I thought we got our act together second half, we just couldn’t hit the target.
“We have dominated the game but we have not controlled it.”
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Man United vs Crystal Palace: Solskjaer reacts to 2-1 loss