ESPN reported last month that United were set to pay £64 million ($82m) plus another £8m ($10m) in add-ons for the 20-year-old striker.
"It is no secret that I have been a fan of this great club since I was a small boy, and I dreamed of walking out at Old Trafford as a Manchester United player," Højlund said in a statement.
"I am incredibly excited by this opportunity to turn that dream into a reality, and I am determined to repay the faith that the club has shown in me. It is still early in my career, but I know that I am ready to make this step up and play with this group of world-class players."
Sources told ESPN that while Ten Hag had previously preferred to add an experienced striker to his squad, the Dutchman has come round to the idea of working with youngster Højlund because of his potential.
"Once I had spoken to the manager, I knew that this environment would be perfect for my development," Højlund added.
"I am relishing the opportunity to work with one of the best coaches in the world. Under his guidance and support, I know that I am capable of achieving great things together with my new team-mates at this special club."
Højlund scored nine goals in 32 appearances for Atalanta in Serie A last season.
United football director John Murtough said: "Throughout the summer we have acted quickly and decisively to secure our primary targets early in the transfer window. This will ensure that Erik and his coaches have the best opportunity to prepare the squad to push for further success in the exciting campaign ahead."