He damaged cartilage in his left knee in a collision with Leighton Baines in a 3-1 home win over Everton.
Manager Jurgen Klopp had said Mane, 25, needed surgery, leaving it "pretty much impossible for him to play again this season".
Liverpool are third in the Premier League and have six games left.
The injury is expected to rule him out for two months.
Mane joined the club for £34m from Southampton last summer and has started all but six of Liverpool's league games this campaign.
Of those, one was won, three were drawn and two were lost.
Klopp, speaking before Saturday's win at Stoke, also said Adam Lallana was "much better but is not in training" as the midfielder continues his recovery from a thigh injury suffered on England duty in March.
Captain Jordan Henderson, who has been out since February, is "in a good way, but I don't know when he can be part of training again", the German added.