Substitute Donny van de Beek scored his first goal for the club on his debut but it was not enough to inspire a comeback.
"You can see we are short and they deserved the points," United manager Ole Gunnar Solskjaer said. "Today was a performance you don't see very often from this group. We should look at ourselves in the mirror. Everyone knows we can perform better than this."
With Van de Beek on the bench, Solskjaer's starting line-up saw David de Gea start ahead of Dean Henderson and included Paul Pogba and Bruno Fernandes, but it was Palace who took control early when Townsend opened the scoring after seven minutes, converting a cross at the back post.
Thereafter, the visitors were most threatening on the counter-attack and De Gea made a smart save to deny Jordan Ayew before the interval, but failed to double their lead until just past the midway point of the second half.
When referee Martin Atkinson agreed with VAR Jonathan Moss that Victor Lindelof had handled Ayew's shot from close range, Palace were awarded a spot kick.
David De Gea saved Ayew's penalty but, after more VAR intervention, it was retaken because the goalkeeper was off his line. Ex-United winger Zaha stepped up to convert from the spot to make it 2-0.
Van de Beek replaced Pogba and made an instant impact with a smart close range finish to close the gap, but Zaha wrapped up the points five minutes later with a fine strike after turning Lindelof.
"In the week of training we did really well, we trained good so my feeling was really positive, Van de Beek told MUTV after the game. "But today we played too slowly, too long with the ball and I think against an opponent like Crystal Palace you need to speed up the game to create something. That was not good enough."
Palace have won their opening two games for the first time in the Premier League era and manager Roy Hodgson was impressed with his side's display at both ends.
"We made certain and kept them playing in front of us and they could not profit from those spaces they are skilful enough to find," he said. "And when we had the ball I thought we looked very dangerous and threatening so I don't think it was a big surprise that we scored three."