The 20-year-old forward has agreed a five-and-a-half year deal but will be loaned back to Palace until the summer.
United will pay £10m up front, with the fee potentially rising to £15m based on Zaha's appearances and success.
Zaha, who joined Palace as a 12-year-old, is expected to win his second England cap against Brazil in February.
Both clubs have refused to comment on a proposed deal, but Sir Alex Ferguson has been heavily involved in the talks, having remained in England as his squad travelled to Qatar on Sunday for a short training camp.
He was named the Football League Young Player of the Year last season and attracted the attention of a number of Premier League clubs.
Zaha, who came through the youth academy at Selhurst Park, has scored 15 goals in 124 appearances for the Eagles since his debut against Cardiff in March 2010.
He is Palace's latest home-grown talent, following in the footsteps of Chelsea's Victor Moses and Southampton's Nathaniel Clyne.
•Born: 10 November 1992
•Club career: Crystal Palace (2009-present)
•Professional debut: v Cardiff (27 March 2010)
•England Under-19 caps: 2
•England Under-21 caps: 5
•England international caps: 1