The Red Devils have been monitoring Powell for some time and have beaten off interest from Everton to sign him.
United have confirmed the youngster has passed a medical and will join the club at the beginning of July.
"We have scouted Nick for months now," said manager Sir Alex Ferguson. "He is an exceptional talent and has been schooled well at Crewe.
"I'm delighted he has chosen to come to United. He has a lot to offer and we are all looking forward to working with him."