The 18-year-old winger has signed a contract until 2025 with the option of a further year.
United reached an agreement with the Serie A club for Diallo on October 5, with just a successful visa application needed before Diallo can travel to Manchester and link up with his new team-mates.
“After waiting since the summer, it is finally a dream come true to complete my move to Manchester United,” Diallo said.
“This is a really exciting squad with so many great players, I can promise that I will work hard every day to learn from them and give everything to develop my game. The coaches have been fantastic since I signed, we have been in touch regularly and they have already given me a lot of great advice.”
Diallo made five appearances for Atalanta — including two this season — and scored one goal, on his debut against Udinese in October 2019.
“I have had time to prepare for this move, both physically and mentally, and I have worked really hard to be ready to make the step up to this amazing club,” Diallo added.
United will pay an initial £19m and a further £18m in performance-related add-ons.
“As a club, we have followed Amad for a number of years and, having watched him myself, I believe he is one of the most exciting young prospects in the game,” Ole Gunnar Solskjaer said.
“Manchester United has such a proud history of developing young players and everything is in place to enable Amad to reach his potential here.
“It will take time for him to adapt, but his speed, vision and fantastic dribbling ability will stand him in good stead to make the transition.”