The 26-year-old clutched his hand in pain as blood flowed from the injury. He was taken off on a stretcher before being transported to hospital.
Galatasaray also confirmed that striker Victor Osimhen suffered a fracture to his right forearm in the first half. The club said on Thursday a decision on whether he will require surgery would be made in the coming days.
Liverpool advanced to the Champions League quarter-finals with a 4-1 aggregate win.