The Portuguese coach will move to Old Trafford from Sporting Lisbon on November 11 and has signed a contract until 2027 to become the club's sixth permanent boss since Sir Alex Ferguson retired in 2013.
Ruud van Nistelrooy will remain as interim manager for United's next three matches before Amorim's first game in charge comes away at Ipswich in the Premier League following the international break on November 24.
A United statement read: "Ruben is one of the most exciting and highly rated young coaches in European football. Highly decorated as both a player and coach, his titles include winning the Primeira Liga twice in Portugal with Sporting CP; the first of which was the club's first title in 19 years. Ruud van Nistelrooy will continue to take charge of the team until Ruben joins."
The club will announce the members of Amorim's backroom staff at a later date and the coach will still be in charge of Sporting for their Champions League match against Manchester City next week.
Amorim has been hugely successful at Sporting and acknowledged that his players would be "disappointed" at his decision to leave the club mid-season and join United.
He said: "Without a doubt, without a doubt. But I don't interfere in the players' lives. Now I talk to the players, because I'm Sporting coach. I want to do well for the group.
"I talked to the board about how we are going to manage the squad. But without a doubt my players will be, if I leave, they will be disappointed in me and sad with me. But it's part of life."