The 20-year-old's impending move, which is due to be finalised at some point next week, was announced today after both clubs reached an agreement on a reported £15 transfer fee.
"I am delighted to be joining Manchester United and ending the recent speculation," Zaha told Palace's official website. "I would like to thank the owners at Palace for making the move possible, however my focus for the next five months will be solely at Crystal Palace.
"I have been at Palace for more than 10 years, I will always love the club and I want to help get them get back to the Premier League, particularly for the fans who have always been so good to me."
Palace co-chairman Steve Parish added: "I feel this is a great deal for the club and the player. We would like to thank Wilf for all he has done for the club and we all wish him well."
Zaha will join up with his new teammates in the summer after completing the season on loan at Selhurst Park.