Arne Slot's side had squandered the lead given to them by Dominik Szoboszlai's second goal this season as Adam Armstrong levelled before the break at St Mary's.
Mateus Fernandes put bottom of the table Southampton ahead after the interval before Salah sparred Liverpool's blushes.
Salah coolly netted the equaliser and then held his nerve to stroke home the late penalty that gave him 12 goals in all competitions this season and six in his last five league appearances.
Thanks to Salah's late heroics, Liverpool have left second placed Manchester City trailing in their wake after the champions' dismal 4-0 defeat against Tottenham on Saturday.
Slot has made a remarkable start to his time in charge of the Reds, winning 16 of his first 18 games in all competitions.
The former Feyenoord manager's deft touch has established Liverpool as favourites for a first Premier League title since 2020 in his maiden season at Anfield after replacing Jurgen Klopp.
Liverpool were below their best against Southampton but the gritty victory was the ideal start to a pivotal period.
The Reds host Real Madrid in the Champions League on Wednesday, before Manchester City head to Anfield next weekend.
Liverpool have won 10 of their first 12 league matches, with only two previous teams in the competition's history failing to lift the title after reaching that mark.
On a wet and windy afternoon on the south-coast, Liverpool threatened to blow Southampton away as Salah unleashed a curling effort that forced Alex McCarthy into a sprawling save.
Salah volleyed over from Andrew Robertson's cross and went close again moments later, taking a touch at the far post before drilling a low strike that McCarthy repelled at full-stretch.
Szoboszlai was next to test the over-worked McCarthy with a long-range blast, before Cody Gakpo almost beat the Southampton keeper from the edge of the area.
Salah to the rescue
Southampton manager Russell Martin has been criticised for sticking to his philosophy of playing out from the back despite some costly mistakes from his defenders.
Once again, Martin's men were punished for taking an unnecessary risk in their own area as Liverpool took the lead in the 30th minute.
McCarthy needlessly rolled the ball to Fernandes, who was immediately put under pressure by Curtis Jones, prompting a panicked clearance from Flynn Downes.
Szoboszlai seized on the miscued pass and slotted a composed finish past McCarthy from 18 yards.
Out-played for much of the half, Southampton snatched a controversial equaliser in the 42nd minute.
Virgil van Dijk carelessly conceded possession to Tyler Dibling and Robertson responded by tripping the winger on the edge of the area.
Referee Sam Barrott pointed to the spot even though the foul appeared to have taken place just outside the box, with VAR supporting his decision.
Armstrong's penalty was saved by Caoimhin Kelleher, but the Southampton forward reacted quickest to smash the rebound past the Liverpool keeper.
Liverpool had lost their momentum and they were hit with a sucker punch in the 56th minute.
Dibling started the devastating break with a superb long pass that Armstrong surging towards the Liverpool area, where he checked back and found the unmarked Fernandes, who guided an inch-perfect shot past Kelleher.
But Salah hauled Liverpool level nine minutes later after another McCarthy howler.
Salah's well-timed run beat the Southampton offside trap and McCarthy's ill-judged decision to race off his line allowed the Egyptian to meet Ryan Gravenberch's pass with a predatory finish.
Liverpool could scent victory and they were handed the points with seven minutes left.
Yukinari Sugawara blocked Salah's cross with his arm and the Egyptian stepped up to drive the penalty past McCarthy before whipping off his shirt in a delirious celebration.
