The versatile 20-year-old has signed a contract with United until June 2030, with the option of an additional year, becoming the second signing for head coach Ruben Amorim.
Dorgu, who can be deployed as a wing-back and outright winger, will reportedly cost the Red Devils an initial €30 million (£25.2m) and €5 million (£4.2m) in potential add-ons.
"I am incredibly proud to be able to call myself a Manchester United player; this is a very special day for my whole family," Dorgu said.
"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."
Amorim earmarked a wide defender on his list of priorities to address in January with Dorgu identified as a prime fit for the Portuguese's 3-4-2-1 system.
Dorgu started his career in Denmark at Nordsjaelland, progressing through their famed youth set-up before moving to Italy to play for Lecce.
He made his debut for the club in the 2023/24 season and established himself as an ever-present in the team since.
"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," United technical director Jason Wilcox said.
"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 bought 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."
Dorgu made 23 appearances in all competitions for Lecce this term, scoring three goals including two game-winners.
Dorgu is already a full Denmark international, earning four caps and scoring one goal since making his debut in September.