Both teams saw goals disallowed in an eventful stalemate which saw the Eagles leave with their first ever Premier League point from a visit to Old Trafford.
Palace did not take long to settle into the game and began to give as good as they got, with former United winger Wilfried Zaha shooting over the crossbar and narrowly wide within the space of a couple of minutes.
Zaha also played a role in supplying Patrick van Aanholt with the best chance of the game, teeing up the unmarked Dutchman 15 yards out but he scuffed his right-footed shot wide at the end of a flowing counter.
The hosts did not test Wayne Hennessey until the half-hour mark, and then did so twice in a matter of moments. The Palace goalkeeper reacted well to get down to Lingard's effort from Ashley Young's low cross, and then was back on his feet quickly to save from Anthony Martial's header.
Chris Smalling also came close for United when heading wide from a difficult angle at the back post on 40 minutes, but it was the Eagles who may have considered themselves unlucky not to be ahead at the interval after having a goal ruled out, although the assistant referee was vindicated in his decision.
When a free kick was drilled in from the left byline, Cheikhou Kouyate was narrowly offside and although it was Victor Lindelof who headed the ball into his own net, the Palace midfielder was undoubtedly interferring with play.
It was United's turn to have a goal disallowed 12 minutes after the restart, with the flag denying Romelu Lukaku a first league goal since September when he tapped home the rebound to Lingard's shot from an offside position.
Jose Mourinho tried to inspire an improvement in the final quarter of the game by calling for Marcus Rashford, Marouane Fellani and Alexis Sanchez from the bench, but none of the Portuguese's substitutes could rescue a disappointing United display.
David de Gea was called upon to make a big double save and ensure things would not get worse for the Red Devils, pushing away Kouyate's 20-yard drive and then blocking from Van Aanholt's close-range follow up on 63 minutes.
The better chances continued to fall Palace's way and Andros Townsend was guilty of missing a golden opportunity, although it was all of his own creation as he dribbled through four defenders before spooning a shot horribly wide when one-on-one with De Gea.
Martial could not continue his goalscoring streak, as he dragged a 79th-minute shot into the side-netting, and when United did engineer a clear-cut chance, Hennessey kept them at bay with a diving stop from Lukaku's guided header.
Top-four rivals Tottenham Hotspur and Chelsea face off in a blockbuster London derby this evening, while Arsenal entertain high-flying Bournemouth tomorrow when the gap to the Champions League places could be as much as seven points to United.
MAN UTD (4-2-3-1): De Gea; Darmian, Lindelof, Smalling, Young; Matic, Pogba (Sanchez, 68'); Mata (Rashford, 60'), Martial, Lingard (Fellaini, 60'); Lukaku
PALACE (4-3-3): Hennessey; Wan-Bissaka, Tomkins, Sakho, Van Aanholt; Milivojevic, McArthur, Kouyate; Meyer, Townsend (Schlupp, 88'), Zaha (Ayew, 90')