The Reds went in front with 10 minutes elapsed at Anfield when Xherdan Shaqiri’s strike deflected off Shane Long and against the shins of Wesley Hoedt and over his own goal line.
Joel Matip then marked his maiden start of the season by netting his first goal since last November when he diverted a corner into the back of the net with his head.
Salah added the game’s third goal, slotting in from close range after Shaqiri’s stunning 25-yard free-kick had crashed back off the underside of the crossbar.
It meant Liverpool returned to the summit of the fledgling Premier League table, with Chelsea not in action until Sunday at West Ham United, as their outstanding start to 2018-19 continued.