The versatile youngster has signed a long-term contract (until June 2030) with United, becoming the second new addition under head coach Ruben Amorim at Old Trafford.
Dorgu, who is of Nigerian descent, can be deployed as a wing-back and outright winger and will reportedly cost the Red Devils an initial €30 million and €5 million in potential add-ons.
Amorim earmarked a wide defender on his list of priorities to address in January with the Dane identified as a prime fit for the Portuguese's 3-4-2-1 system.
"I cannot wait to work with Ruben Amorim; his vision for this team and the future of the club is incredibly exciting," Dorgu said in a statement.
"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."
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.
Dorgu made 23 appearances in all competitions for Lecce this campaign, 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.