The former Crystal Palace winger will link up with his new team for the first time next month as he looks to hit the ground running in his debut Premier League season
Zaha joined United from Palace in January before returning to Selhurst Park on loan and has been one of the most sought-after talents in the country after helping fire his boyhood club into the Premier League via the play-offs.
David Moyes is yet to decide whether to make the 20-year-old part of his first team plans or loan him out, with West Ham registering their interest in taking him on loan.
But Freedman, under whom Zaha established himself as one of the most exciting talents in the Championship, has backed the Ivory Coast-born winger to make his mark on Old Trafford if he is given the time to do so.
"It took him eight games to establish himself in the Palace first team and eight to 10 games later he was the best player," the Scot told reporters.
"I have no doubt he will take time to settle in but when he does he will show the talent that he has.
"I know his character and he won’t be fazed, David Moyes will see that day-in and day-out."
Zaha scored six goals and provided 10 assists for the Eagles last season, before scoring twice in the play-off semi-final against Brighton and winning the deciding penalty in the final against Watford.