Zaha will join United formally at the end of this campaign.
He told The Sun: "It is difficult to get your head around how quickly this has all happened. Sometimes, I sit here in this very room with my brother Kel and we talk to each other in amazement. How has this happened all so quickly?
"All I can do is keep my head down. I have to try to cope with this success, while doing my best for Palace until the end of the season.
"Back in August the goal was to have a good season. I never thought I'd end the season a Manchester United player.
"Then things started to happen with Palace. While having a meal in November to celebrate my birthday, I got the call-up to the England seniors. I was thinking 'this is crazy'.
"I was part of the squad which played in Sweden. The England team all said 'hello' to me.
"Steven Gerrard came up to me and gave me some advice. He settled me down which was very nice of him as I was a bag of nerves.