The Reds were behind inside a minute when Harry Kane stooped to head home and Spurs maintained that advantage until the interval.
Skipper Jordan Henderson scored his first Anfield goal in four years to level seven minutes into the second half before Salah’s spot kick 15 minutes from time, after Serge Aurier’s clumsy challenge on Sadio Mane, ultimately secured the points.
Jurgen Klopp’s side have become just the fourth team in Premier League history to take 28 points from the first 30 on offer, with all three teams that have achieved the feat previously having gone on to claim the title.
Skipper Henderson will be relieved to have scored the leveller as he was culpable in the early setback. He was robbed in midfield by Harry Winks and Spurs worked the ball to Heung-Min Son.
The South Korean’s shot was deflected onto the crossbar and, with Alisson stranded, Kane reacted quickest to head the ball back into the net and stun Anfield.
Liverpool had to gather themselves after that early setback but they soon took control of the game.
Spurs goalkeeper Paulo Gazzaniga was a busy man in the first half. He had to be alert to deny Salah and Roberto Firmino in quick succession midway through, while Virgil van Dijk had a point-blank header pushed over the bar.
Trent Alexander-Arnold then powered a shot on target from 25 yards and again the Argentine was again equal to it.
Spurs carried their threat too, with Christian Eriksen’s cross just evading Dele Alli in one of their rare forays into the Liverpool penalty box.
Liverpool were on terms early in the second half. Fabinho lofted a ball toward Firmino and although Rose intervened, he merely teed it up for Henderson to poke his shot across goal into the net.
That was Liverpool’s cue to lay siege on Tottenham, Mauricio Pochettino’s men forced back deep into their own half as the game wore on.
The Reds pushed hard but clear-cut chances were at a premium before Aurier gave them some assistance. Mane chased a long ball into the area but it looked as if the Tottenham defender would outrace him.
The Senegalese refused to give up and he got between Aurier and Toby Alderweireld before the latter clumsily felled him. Salah went for power from the spot, driving low to the left of Gazzaniga.
Trailing, Tottenham finally came out to play and they had the ball in the net through Son only for an offside flag to intervene.
Danny Rose had the final chance for the visitors but he shot wildly over in stoppage time, while Klopp’s only concern will be the fact Salah was subbed late on for the second home game in a row after a late knock.