Manchester City's 3-1 win at Crystal Palace earlier on Saturday piled pressure on the Reds and they passed the test thanks to two second-half goals in almost as many minutes.
Star man Salah rose to an emotionally charged occasion with a thumping 25-yard effort that whistled into the top-left corner and doubled the home side's lead following Sadio Mane's headed opener.
The victory, Liverpool's first against Chelsea at Anfield in seven Premier League meetings, restored a two-point lead at the top for Jurgen Klopp's men and exorcised the demons of the infamous 2-0 loss in this fixture that shattered their 2013-14 title hopes.
Salah, a Chelsea starter here five years ago, volleyed the earliest chance straight at Kepa Arrizabalaga.
Eden Hazard and Willian issued warnings on the counter before Mane, provider of Salah's opportunity, shot wide from the Egypt star's cut-back.
Chelsea encountered their first real disruption when a knee problem forced Antonio Rudiger to the bench before the break.
Two greater blows followed after the restart. Mane arrived at the far post to nod in Jordan Henderson's cute cross in the 51st minute and, moments later, Salah cut inside and sent Anfield ecstasy.
Hazard had an excellent pair of chances to hush the electric atmosphere, but the Belgian hit the base of the left post following Emerson Palmieri's ball over the top and then found Alisson from Willian's right-sided delivery.
Maurizio Sarri's side otherwise had precious few opportunities as their grip on fourth weakened.
Reds clear psychological hurdle
So much of the build-up to this encounter centred on the April 2014 meeting between these teams - and Steven Gerrard's ill-fated slip - that derailed Liverpool's bid for a maiden Premier League title.
But where Brendan Rodgers' side floundered, the 2018-19 vintage are accelerating towards the finish line. They have won seven successive games in all competitions.
Fabinho an essential fixture in midfield
A peripheral figure during the early stages of his debut Anfield season, Fabinho is now entrenched in Liverpool's strongest XI.
The long-striding Brazil international shielded his defence with distinction in this match, making four tackles, and moved the ball around with purpose.
