Warriors said that Kellock was "expected to be out of action for several months" after the planned operation, which was carried out by Mr Umberto Fazzi at BMI Ross Hall Hospital in Glasgow.
Kellock, 32, was not selected for Scotland's summer tour and played through his shoulder problem as Glasgow made it all the way to the RaboDirect PRO12 final for the first time. The new season starts in September.
"A number of other players are currently being assessed by the club's medical team following Saturday's RaboDirect PRO12 final in Dublin," said Warriors in a statement.
"Glasgow Warriors would like to thank the Leinster team doctor Professor John Ryan and all the staff in the A&E department of St Vincent's Hospital in Dublin for all their help following the game."