Jurgen Klopp's side had surrendered pole position a few hours earlier when Manchester City won 3-1 at Crystal Palace, piling pressure on the Reds to respond as the title race heats up.
Liverpool rose to the challenge as Sadio Mane's opener five minutes after the break and a blistering long-range drive from Salah two minutes later moved them two points clear of second-placed City on a raucous afternoon at Anfield.
City have five games left compared to just four for Liverpool, leaving the destiny of the title in the hands of the champions.
But there is a growing feeling at Anfield that this will be the season Liverpool finally end their 29-year wait to be crowned kings of English football.
Chelsea posed the last major threat to Liverpool's title bid, yet the leaders dealt with the challenge impressively to mark Klopp's 200th game as Reds manager in style.
With City having difficult league fixtures looming against Tottenham and Manchester United, Klopp's men are convinced they can hold onto pole position.
Their run-in is undoubtedly less daunting than City's, with Cardiff, Huddersfield and Newcastle on the schedule before a potential title party against Wolves at Anfield.
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For fourth-placed Chelsea, the loss was a blow to their bid for Champions League football next season and they will drop to fifth if Arsenal win at Watford on Monday.
On the eve of the 30th anniversary of the Hillsborough disaster that killed 96 Liverpool fans and in the week that club legend Tommy Smith passed away, there was even more emotion at Anfield than usual.
After a minute's silence featuring mosaics reading "30 years" and "96" held up by fans around the ground, Liverpool's focus was trained on avenging one of the more painful defeats in the club's recent history.
Steven Gerrard's infamous slip in 2014 led to a defeat against Chelsea that effectively gifted the title to Manchester City when Liverpool had looked odds-on to lift the trophy.
But Klopp insisted his players wouldn't be haunted by the ghosts of that collapse and they were true to his word.
Liverpool threatened an early goal when Mane whipped a pin-point cross towards Salah and he fired a volley that Chelsea keeper Kepa Arrizabalaga held well.
Chelsea were unbeaten in their last eight matches at Anfield going into Sunday's clash and showed signs of troubling Liverpool when Eden Hazard twisted away from Joel Matip for a low shot that forced Alisson Becker into action.
David Luiz launched a dangerous Chelsea counter-attack with a pass that sent Willian racing away, but the Brazilian's miscued shot drifted wide.
Those escapes shook Liverpool out of their brief lull and Salah got behind Emerson to cross for Mane to smash just wide.
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Hampered by a first-half injury to Antonio Rudiger, Chelsea creaked at the back and Liverpool took full advantage to break the deadlock in the 51st minute.
When Salah scampered onto Roberto Firmino's flick in the Chelsea area, Blues defender Emerson could only prod the ball to the influential Jordan Henderson and his deft cross reached the unmarked Mane, who headed home at the far post.
Any Liverpool nerves had been wiped away by jubilant celebrations and soon after Salah blew the roof off Anfield with a moment of pure genius.
Taking possession wide on the right flank, Salah cut inside and glided away from Emerson before unleashing a sumptuous 25-yard drive that arrowed past Kepa into the far corner.
It was a sweet moment of vindication for the Egypt forward after a group of Chelsea fans were filmed chanting "Salah is a bomber" before their team's Europa League tie at Slavia Prague on Thursday.
And Salah, a practising Muslim, celebrated his 22nd goal of the season by adopting a prayer position in front of the Kop.
Hazard should have reduced the deficit, but twice the Belgian couldn't find the net from close-range as Liverpool moved a step closer to their holy grail.
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