Damien Delaney set United on their way when he inadvertently turned Matteo Darmian's cross into his own net.
Julian Speroni kept Palace in the game with a series of fine saves before Darmian doubled the lead by lashing home from 20 yards.
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Arsenal, who have a game in hand, entertain West Brom on Thursday.
Darmian's first goal for United sealed a highly satisfactory win for Louis van Gaal's side on a night thousands of United fans decided to stay away.