Financial details about the transfer were not disclosed by either club, but British media reported that the deal was worth around 75 million euros plus 20 million euros in potential add-ons.
"Today I say goodbye to this wonderful club with many emotions. They were two very special years. During this time, I grew and learned a lot as a player and as a person," Alvarez said on Instagram.
"I will always be very proud to have been part of this family, Manchester City will always have a special place in my heart. I wish you every success in the future and will continue to support the club from wherever I am."
The Argentina international's exit marks a record sale price for City, eclipsing the 50 million pounds Chelsea paid for winger Raheem Sterling in 2022.
Alvarez joined City from Argentine club River Plate in 2022 for 14 million pounds and has scored 36 goals in 103 appearances for manager Pep Guardiola's side.
He has won the Premier League twice and the Champions League, FA Cup, Super Cup and Club World Cup all once. He also won the 2022 World Cup with Argentina and the Copa America earlier this year.
Atletico manager Diego Simeone had been on the lookout for a centre forward since Alvaro Morata's move to AC Milan last month.
