The 23-year-old Uruguay international moves for a £50.5m fee and has penned a five-year deal with the option for a sixth.
"It is an incredible feeling to join a club of this magnitude; one that is admired all around the world," said Ugarte.
"Manchester United is an ambitious club and I am an ambitious player. I know how incredible United fans are and I cannot wait to experience Old Trafford.
"I'm someone who is so determined to succeed; I will sacrifice and give everything for my team-mates. Together we will fight to win trophies and reach the level where this club needs to be."
United sporting director Dan Ashworth said: "Signing Manuel was another of our primary targets for this summer.
"He is among the very best ball-winning midfield players in the world and has an excellent record at both club and international level. His qualities, experience and passion will be a great complement to our strong group of midfielders."