"Everything went well, and I'll soon begin recovery," Aguero said on Twitter with a photograph of himself lying in a hospital bed.
City manager Pep Guardiola told reporters earlier that the Argentine would have the operation on his left knee on Thursday.
The club's record goalscorer limped off with the injury in Monday's 5-0 Premier League home win over Burnley.
Guardiola said after the game he feared the 32-year-old could miss the rest of the league season which ends in July.
"After the surgery... he will tell us with more accuracy when he can come back," he told reporters ahead of Thursday's game at Chelsea.
Aguero flew to Barcelona on Tuesday to see specialist doctor Ramon Cugat and City will be hoping the player might return in time for the Champions League when it resumes in August.
City lead 2-1 against Real Madrid after the away leg of their last-16 match and are hoping to reach the 'Final Eight' of the competition which will be held in Lisbon.
Guardiola said City had plenty of options, with Gabriel Jesus the only main striker fit and ready but Ilkay Gundogan and Raheem Sterling are able to slot into the role.
The manager indicated defender John Stones was close to a return after an ankle injury but was unlikely to face Chelsea.
"He is on the pitch training and getting better," said the Spaniard. "Hopefully in the next few days he can play in the team."
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