The Premier League club reached an agreement to sign the England winger from Borussia Dortmund last month, and the 21-year-old successfully underwent a medical following his participation at Euro 2020. He has signed a five-year deal at Old Trafford.
United are understood to have paid €85m (£72.9m) for the England international, making him the fourth most expensive signing in their history.
“I’ll always be grateful to Dortmund for giving me the opportunity to play first team football, although I always knew that I would return to England one day,” Sancho said.
“The chance to join Manchester United is a dream come true and I just cannot wait to perform in the Premier League. This is a young and exciting squad and I know, together, we can develop into something special to bring the success that the fans deserve. I am looking forward to working with the Manager and his coaching team to further develop my game.”
Sancho scored 50 goals in 137 appearances for Dortmund during his four seasons at the club, establishing himself as one of the most exciting talents in European football following his transfer from Manchester City in 2017.
He made three appearances for England at Euro 2020, including a late substitute appearance in the final defeat to Italy in which he missed a penalty in the shootout.
Sancho was one of three England players, along with new Manchester United team-mate Marcus Rashford and Bukayo Saka, to receive racist abuse following the final.
The signing represents a key step for United under manager Ole Gunnar Solskjaer, who said the club must “strengthen” this summer in order to catch up with Premier League champions and rivals Manchester City.
“Jadon epitomises the type of player I want to bring to the club, he is a forward player in the best traditions of Manchester United. He will form an integral part of my squad for years to come and we look forward to seeing him blossom. His goals and assists records speak for themselves and he will also bring tremendous pace, flair and creativity to the team,” Solskjaer said.
“Old Trafford will give him the platform he needs to release his untapped talent and perform at the highest level. For a player of his age, Jadon has already achieved a great deal and showed the courage to go and prove himself abroad. We all look forward to welcoming Jadon to the squad as we prepare for the new campaign.”
Sancho, who United tried to sign last year, will join up with the squad following his summer break due to his involvement in the Euros.