The Reds went behind inside the opening ten minutes as Jetro Willems sent a powerful shot across Adrian and into the back of the net to give the visitors a surprise lead.
Liverpool looked lacklustre in the opening stages but grew into the game as the first half wore on, getting their reward just before the half-hour mark as Mane excellently finished from inside the box after being found by Andy Robertson.
He was on target again five minutes before the break as he took advantage of Martin Dubravka's fumble and poked the ball home after being initially played in by substitute Roberto Firmino, who put in a man-of-the-match performance after coming on.
Firmino linked up with an otherwise quiet Salah in the second-half, playing the ball back to the charging Egyptian with an excellent flick, with Salah finding the back of the net after making his way into the box.
The Reds have now won 14 league games in a row and maintain their flawless start to the current campaign, topping the Premier League table with maximum points after five games.
Slide through our gallery below for all the best photos from Saturday.