The deal is worth an initial €30m, which will rise to €35m in bonuses.
Dorgu has penned a deal with United to 2030.
He said, "I am incredibly proud to be able to call myself a Manchester United player - this is a very special day for my whole family.
"I cannot wait to work with Ruben Amorim. His vision for this team and the future of the club is incredibly exciting.
"There is a clear plan set out for my development, and I feel that Manchester United is the perfect place to fulfil my potential and complete my huge ambitions."
Jason Wilcox, Manchester United's technical director, said: “Patrick is a really exciting talent; his strong defensive and attacking attributes, adaptability and work-rate will make him a key part of Ruben Amorim’s squad.
“Patrick has had a superb start to his career, and at the age of just 20, we know he will continue to develop under the guidance of our excellent coaching team.
“Patrick is the latest exciting young player that we have brought into the club. We believe that each of these talents has world-class potential and the ability to play a key role in achieving the success that everyone at Manchester United is striving for.”