Ole Gunnar Solskjaer's men earned a 3-1 victory, despite being without nine players due to injury, and Marcus Rashford was only fit enough to make the bench after hurting his ankle against Liverpool.
Romelu Lukaku opened the scoring in the first half with a well-placed finish to cap off a positive surge forward by Luke Shaw.
After the break, the big Belgian did the damage again by turning to slam the ball home after headers by centre-backs Chris Smalling and Victor Lindelof.
Joel Ward's diving header reduced the arrears for Palace, only for Ashley Young to beat Ward on the line with a fierce drive, after being set up by Paul Pogba.