There was a paucity of quality in misty conditions at Selhurst Park as Ole Gunnar Solskaer's men handed Manchester City another boost in the Premier League title race.
City's 4-1 win over Wolves on Wednesday stretched the gap at the top to 15 points, but United could not respond in kind against a Palace side who beat them 3-1 at Old Trafford in the reverse fixture.
The Red Devils failed even to have a shot between the 27th and 81st minute and have now gone three games without scoring a goal, having recently been the leading scorers in England's top flight this term.
United started scrappily but twice came close to an opening goal inside the first 20 minutes, Nemanja Matic's deflected shot brilliantly saved by Vicente Guaita before Marcus Rashford skewed a clear chance wide.
Aaron Wan-Bissaka's cross nearly provided Edinson Cavani with a tap-in, but it was mostly a lifeless first-half display from the United attack, in keeping with their previous back-to-back goalless draws.
There was little change early in the second half and Palace came close to snatching the lead through a half-volley from Andros Townsend that bounced wide.
Substitute Dan James missed the ball when presented with a free header from a Luke Shaw cross, and Mason Greenwood fired over from the edge of the box, but that was as close as the visitors came.
Patrick van Aanholt should have won the match when he was found by Luka Milivojevic, but Dean Henderson, in for the absent David de Gea, saved superbly at the feet of the left-back.