Borussia Dortmund had previously confirmed the agreement, stating on July 1 that the transfer fee is "fixed" at €85 million ($99.98m). United also confirmed that the deal had been initially agreed to, subject to a medical after England's participation at Euro 2020.
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."
The deal officially brings to an end a transfer saga which has gone on for nearly 18 months.
United manager Ole Gunnar Solskjaer said: "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. 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."
United tried to sign Sancho last summer but failed to agree a fee with Dortmund. The German side lowered their asking from £108m ($148m) and after seeing at least two bids knocked back United were finally able to reach an agreement this summer.
Sources have told ESPN that Sancho's former club, Manchester City, are due 15% of the profit Dortmund have made on the deal -- expected to be around €11.2m ($13.17m).
Sancho leaves Dortmund after making 137 appearances, scoring 50 goals, in four years in Germany. He was named in the Bundesliga team of the season in 2019 and 2020.
Dortmund CEO Hans-Joachim Watzke said he had hoped to keep Sancho this summer, but that the 21-year-old had told the club he wanted to move to Old Trafford.
"It was the exemplary wish of Jadon," Watzke said. "We would have liked to keep him.
"We reached a general agreement with Manchester United on the transfer of Jadon Sancho, pending full completion, as there are still some formalities to agree on and the medical check to be completed."