The 17-year-old is a product of the Red Devils' academy system and has represented the club at every level.
Gomes, who made his senior debut for United in the final game of last season against Crystal Palace, would have been able to leave for a nominal fee at the end of this campaign.
"Manchester United has always been a big part of my life," the midfielder told the official club website. "I have been with the club since the age of six and it was a proud moment for me when I made my first team debut.
"This club is renowned for giving home grown players opportunities to progress into the first team and that was a major factor in me signing my first professional contract at this great club.
"I am still young but am looking forward to continuing my development with the help of the coaching staff and I will take every opportunity that comes my way."
United academy head Nicky Butt added: "Angel has gone from strength to strength in his development and we are all pleased that he has signed his professional contract with Manchester United.
"He is still very young and has a good aptitude for learning; coupled with his natural talent, we can see that Angel has a very bright future."