Ayew fired Roy Hodgson's men ahead in the 32nd minute after linking up with compatriot Jeffrey Schlupp's header.
The goal ended the Eagles' abysmal 39-year run without a lead at Old Trafford, with David Swindlehurst last managing the feat in 1979.
Daniel James cancelled out Ayew's opener towards the end of the encounter but Patrick van Aanholt secured the maximum points for the visitors with his stoppage time goal.
Ayew was replaced by Christian Benteke in the 75th minute while Andros Townsend came on for Schlupp five minutes later.
The win was Crystal Palace's first for the season and they climbed to 10th in the table with four points after three matches.
Roy Hodgson's side battle Colchester United in Tuesday's League Cup outing before hosting Aston Villa for their next Premier League game on August 31.