With nearest rival Manchester City playing at Tottenham on Sunday, Juergen Klopp's side turned the screw even further with second-half goals from Alex Oxlade-Chamberlain - against his former club - Jordan Henderson and then Salah.
Seven more wins will guarantee winning the league and Liverpool has now won 24 of its 25 matches, with nine clean sheets in 10 games in the process. Salah, Georginio Wijnaldum, Oxlade-Chamberlain and Trent Alexander-Arnold all had shots blocked or saved but the biggest chance of the opening 45 minutes fell to Virgil Van Dijk, whose first touch killed Salah's cross before his backheeled shot was saved.
Southampton goalkeeper Alex McCarthy blocked and saved the follow-up from Roberto Firmino, although the Brazil international felt he should have had a penalty in the next phase of the attack when Shane Long appeared to pull him down almost on the goalline.
Referee Kevin Friend waved play on and VAR official Simon Hooper concurred. It was easy to see, however, why two-thirds of Saints' points this season have come away from home as they pressed high and worked tirelessly to nullify Liverpool's threat, both short and long, while waiting to pounce on any mistake.
They were presented with plenty of those in the first half, with Joe Gomez particularly slow in responding to threats and Firmino playing a 30-yard back pass to Danny Ings in the penalty area. The former Liverpool striker's shot was blocked by the leg of teammate Long.
Since Southampton's 9-0 home defeat to Leicester on Oct. 25, only Leicester, Liverpool, Manchester City and Manchester United have won more points than Ralph Hassenhuttl's side (23 points from 14 matches) and the fact they out-shot their hosts 10 to eight in the first half was an indicator of that form.
But Southampton was left to rue a number of opportunities in which it failed to properly test Alisson Becker, and teams visiting Anfield rarely get a second chance. The turning point came early in the second half. Ings thought he should have had a penalty after stumbling through a challenge by Fabinho, who was making his first league start after injury since Nov. 23.
The visitors' error was not to react quickly enough as Liverpool, as many teams have discovered to their considerable cost over the last couple of seasons, is at its most dangerous when opponents are on the offensive.
A counterattack down the left eventually saw Firmino pull the ball back to Oxlade-Chamberlain, who cut inside and drilled home a right-footed shot which had McCarthy rooted to the spot. An energized Liverpool stepped things up and Salah had the ball in the net, only to be denied by an offside flag, although Ings continued to create problems until his substitution to a warm round of applause in the 70th minute.
By that time, the visitors were 2-0 down after Henderson calmly slotted the ball home on the hour mark. Salah was not to be denied and, when Alisson picked out Henderson on the right wing, he slid in a pass for his teammate to dink over McCarthy, with the Egypt international adding his second in a goalmouth scramble.
The win brought up 100 points from the last 102 available. With Liverpool unbeaten in the league in 394 days - and just less than three months short of three years at home in the competition - no wonder all four sides of Anfield were belting out “We're going to win the league” at the end.
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