Almost five years since Liverpool's title chances were ruined by a 2-0 defeat in the same fixture, the hosts kept themselves in the hunt for a first league title in 29 years with two goals in the space of two minutes that saw Anfield erupt.
After a nervy first half in which both sides had chances, Liverpool emerged from the break with added purpose and took the lead via Sadio Mane's header.
There was a huge sense of relief inside the ground, but that became a deafening roar when Salah smashed a left-footed angled drive into the top right corner from 25 yards for his 19th Premier League goal of the season.
Chelsea, who had been unbeaten at Anfield since 2012, almost hit back six minutes later when Eden Hazard shot against the post before the Belgian saw another effort saved by goalkeeper Alisson.
But Liverpool could have extended their lead before an exultant Kop greeted the final whistle with roars of delight and Jurgen Klopp punched the air after his 200th game in charge.
Manchester City, 3-1 winners at Crystal Palace in Sunday's other match, still have a game in hand on the Reds, but with games at home to Tottenham and away to Manchester United among their five remaining fixtures, their task looks tougher than the one facing Liverpool, who meet Cardiff, Huddersfield, Newcastle and Wolves.
Chelsea remain fourth, a point behind Tottenham, who have a game in hand on their London rivals. Maurizio Sarri's side could also be overtaken by sixth-placed Arsenal, who are three points behind them, when they face Watford on Monday.
BBCLast modified on Monday, 15 April 2019