Five days after Paul Pogba had a penalty saved in a 1-1 draw at Wolverhampton Wanderers, Marcus Rashford regained spot-kick duties but struck his attempt against the post in the 70th minute, keeping United 1-0 down following Jordan Ayew's 32nd-minute goal.
Daniel James curled in a brilliant equalizer in the 89th minute, seemingly earning United a point at Old Trafford, only for Palace left back Patrick van Aanholt to squeeze a shot under the body of goalkeeper David de Gea at the end of a counter-attack in the second minute of stoppage time.
Palace ended its scoring drought this season to earn a first win from its opening three games.
United picked up four points from its first two games but put in its worst performance of the season.
"The first half was poor," United manager Ole Gunnar Solskjaer said. "In the first half-hour you're feeling you're controlling it and maybe it seems too easy. We didn't really have that zip and urgency about us and they scored with the first chance. Just a long kick, it should never happen, that's poor defending. But in the second half, I thought we were excellent. We were pressing, pushing, creating chances, just not scoring. We didn't have that Midas touch today."