The 26-year-old will be monitored while the 'Reds' prepare for their trip to Huddersfield on Sunday.
Mane suffered the injury while playing for Senegal on international duty and it was clear the winger had broken his thumb after images emerged of him wearing a bandage in a hospital.
Liverpool have now announced: "Sadio Mane today underwent a procedure on a hand injury sustained while on international duty with Senegal.
"The surgery, conducted at a north-west hospital and supervised by club medics, was successful.
"Mane's recovery will be monitored over the next couple of days ahead of the Reds' return to action at Huddersfield Town on Saturday."
Jurgen Klopp's side have several ongoing injury issues, with Mohamed Salah and Naby Keita both doubts for Saturday and with Virgil Van Dijk having worsened a rib issue while playing for the Netherlands.