Egypt international Mohamed Salah scored twice as Liverpool breezed through Red Star Belgrade to move to the top of Champions League Group C.
Roberto Firmino fired the Reds in front after 20 minutes before Salah got his first just before the break then added a second to reach a half-century of Liverpool goals.
Mane then slotted a fourth goal with 10 minutes to go, having missed another penalty just moments before.
Xherdan Shaqiri may not have scored, but he was influential for the hosts as his excellent through ball to Robertson helped set up Firmino’s opener.
Elsewhere, Tottenham played out a 2-2 draw with hosts PSV in Eindhoven, with manager Mauricio Pochettino saying that the club’s chances of reaching the knockout stages are “nearly over”.
“I think it’s easy to judge the game,” Pochettino said. “If you didn’t win following what happened then you don’t deserve to be in the Champions League.
“It was a situation where we needed to think a little bit, because in this type of game, at 2-1 you must kill the game. If not, the opponent is alive.”