The 21-year-old Argentinian had been pushing for a move and becomes the Blues' ninth arrival of the summer.
He told the club's website: "It's an incredible moment for my family and I to join this great club.
"I can’t wait to get started. I watched the Club World Cup and to join the world champions is special – we’re the best team in the world! It’s amazing to be here and I’m very happy."
Garnacho has been a regular in the Man Utd side for the past two seasons and made 93 Premier League appearences, scoring 16 times, including a bicycle kick against Everton that won the 2024 FIFA Puskas Award.
The deal is the fourth largest sale in Man Utd's history after Cristiano Ronaldo, Romelu Lukaku and Angel Di Maria, and the club's largest sale of an academy graduate player.
Garnacho, who has signed until 2032, was spotted at Stamford Bridge earlier on Saturday as Chelsea beat Fulham 2-0.