It was far from a straight-forward victory for the Reds as Chris Wilder's side caused them plenty of trouble at Anfield.
The visitors went ahead early in the first half after a dubious penalty decision, with Sander Berge stepping up to convert the spot-kick.
Mohamed Salah scored a would-be goal-of-the-season contender with a fantastic touch and finish after a lofted pass by Trent Alexander-Arnold. The strike was ruled out for offside by VAR, but it didn't take long for the Reds to strike again.
Mane's cross was met by Diogo Jota, who got ahead of his marker inside the box to head the ball past Ramsdale and secure a hard-fought victory for Liverpool.
Slide through our gallery below for all the best snaps from Saturday.