Early in the second-half, Martial went down under a challenge from Martin Kelly inside the area but did not earn a penalty.
United boss Ole Gunnar Solskjaer spoke about United's problems with penalties, though fans were not impressed with him calling for his side to push for more penalty chances.
Several of Pogba's team-mates - including Rashford - criticised the abuse, while England women's manager Phil Neville called for footballers to "boycott" social media.
On a day of frustration at Old Trafford, Palace scored with two of their three shots on target.
But Solskjaer would do well to look very closely at this 2-1 scoreline, at how his team contrived to hand victory to their opponents as they chased a victor themselves, and give very serious thought to how he can stop it happening again. We created enough to win the game.
United were in control of the game from the first whistle, dominating possession and having all the chances.
Speaking to United's official website, Solskjaer added: "The last two results were disappointments because at Wolverhampton you feel we deserved to win, and today we deserved to win, but you don't always get what you deserve in football". It's unheard of and we need it to stop, it needs to stop.
6 - It wasn't a surprise that the Norwegian named an unchanged XI from the one that drew with Wolves on Monday night and he could do nothing about Lindelof's lapse that allowed Palace to score in the first half.
"If we score them it's a different story".
As much as United pressed forward in the second half, they couldn't conjure the final pass to break down a robust Crystal Palace rearguard, and their own defensive frailties, supposedly reinforced with the addition of Harry Maguire and Aaron Wan-Bissaka (who played against his former club), proved their undoing, particularly with David De Gea no longer being the impenetrable barrier of old. "We were pressing, pushing, creating chances, just not scoring".