The Denmark international has signed a contract until June 2030, with the option of an additional year.
Dorgu, 20, has played 57 games for Lecce, contributing five goals already. The versatile defender has enjoyed an excellent season so far, having won the second-most ground duels in Serie A and attempted the fourth most take-ons of any defender in the league.
He also scored with his first touch on his international debut in September.
Patrick Dorgu 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.