United agreed a deal with Crystal Palace in January for the England international, but allowed Zaha to see out the season at Selhurst Park.
It has been suggested that he could be allowed to return to the Eagles again in 2013/14, allowing him to get useful top flight experience under his belt.
David Moyes has, however, assured the 20-year-old winger that he will get opportunities at Old Trafford and Zaha is happy to stick around and fight for his place.
After figuring in the opening game of United's pre-season tour, he said: "I've grafted to get here so Manchester United is where I want to stay.
"I've moved away from most of my family and friends but it's progress. Everyone understands that.
"I wanted to make the move here because it's what is best for my career."
Zaha is convinced he has what it takes to become the latest in a long line of exciting wingers to grace United's flanks.
He added: "I'm going to do my best to continue those traditions.
"The fans see the wingers we have now and the high standard they bring to the team. I want to meet those high standards as well."