Both sides only managed one shot on target each on what was a foggy night in south London.
United dominated possession throughout but failed to threaten Palace's stubborn and disciplined defence.
Nemanja Matic went close to breaking the deadlock in the first half when his deflected strike seemed destined for the top corner, only to be denied by a brilliant save from Vicente Guaita.
Palace's best chance came in the second stanza through Jordan Ayew, but the Ghanian was denied by Dean Henderson, who did himself no harm in a rare Premier League start.
The Eagles are now 11 points clear of relegation.
Speaking after the game, Ole Gunnar Solskjaer admits his side were 'off' against Palace.
"It's disappointing. We were just a little bit off," said Solskjaer. "If you're not spot on you won't win games. Palace always make it hard for you.
"In the second half we never got going.
"Sometimes it's human nature, sometimes you don't work at your best. Today we didn't. It's been a long season – I'm not blaming that because everyone has the same season. You're playing against a good team who make you play the way they want to.
"We kept giving the ball away. There were times in the first half we built some pressure, had some half decent chnces but kept giving it away. Then you don't build momentum and that's the game you get."
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