After a closely fought first-half, Sadio Mane headed the hosts in front shortly after half-time following a superb chipped cross from Jordan Henderson.
Just two minutes later, Salah picked up the ball from 25-yards and unleashed a thunderbolt into the top corner, leaving Blues stopper Kepa Arrizabalaga with no chance.
Salah’s goal was worthy of winning a title let alone a game, but for now Liverpool will settle for the latter.
A two-point lead over City, winners at Crystal Palace, was restored and proved Jurgen Klopp’s previous assertion that no-one is getting rid of his side in this most enthralling of races.
This was, on paper, the toughest task remaining for the Reds, with games against Cardiff, Huddersfield, Newcastle and Wolves to follow.
The fixture had drawn comparisons with the last time Liverpool were in a title race in 2014, only to be derailed by Chelsea and Jose Mourinho in a match best remembered for Steven Gerrard’s fateful slip, but this was not even a pale imitation of that.
Maurizio Sarri’s side came knowing they had to get something to strengthen their own claims for a top-four spot and, as a result, Chelsea were by no means as defensive as five years ago, looking to hit their hosts on the counter using the talent of Hazard as a false nine.
Too often, though, the Belgium international was left to do it all on his own and he got little change out of Virgil Van Dijk until the hour mark when his side were 2-0 down.
Realistically, however, he should have put his side level as first he beat Alisson Becker in a one-on-one only for his shot to rebound off a post and then fired a first-time shot straight at the goalkeeper from close range seconds later.
It was as manic a couple of minutes as those which saw Liverpool take a lead and quickly double it just after half-time.
Salah nicked the ball off Emerson deep in his own penalty area and Jordan Henderson seized on the mistake to clip a cross to the far post where the unmarked Sadio Mane headed home his 18th Premier League goal of the season.
It was not a great passage of play for the Chelsea left-back as he then stood off Salah enough to encourage the Egyptian to cut in on his left foot and rifle an unstoppable shot into the far top corner.
It was Salah’s first goal in six appearances at home and could not have been better timed. Or more perfectly struck.
Anfield was bouncing at that stage and Roberto Firmino then drilled a low shot just wide as the hosts threatened to run riot.
But just as quickly it changed when Maurizio Sarri brought on Gonzalo Higuain and restored Hazard to his usual position wide on the left.
The Belgian had two chances to score as the game became far more open as, for the first time this season, Klopp’s side looked in danger of bottling it.
Considering how much was at stake the nerves reverberating around Anfield were understandable, although that anxiety at times transferred itself on to the pitch.
For a brief second there was an horrific flashback to Gerrard in 2014 when Andy Robertson slipped on the halfway line, but the ball fell to Higuain, who has considerably less pace than Demba Ba showed five years ago, and Liverpool had plenty of players to cover.