The 20-year-old agreed to join the Red Devils in January from Crystal Palace, but he remained with the Eagles on a loan basis for the remainder of the campaign.
He has now linked up with United and having spent some time training in his new surroundings, Zaha has compared the experience to a "dream".
"It was a bit nerve-wracking at first, coming in and seeing all these players that I've watched over the years. But everyone seems all right so I'm settling in okay," he told MUTV.
"I'm speechless really, I just never thought I'd have the chance to play with the likes of Rooney and to actually be sitting next to Giggs in the changing room. It's just a dream really."
Meanwhile, Zaha has been confirmed as part of the United squad for this summer's pre-season tour of Asia and Australia.