Jurgen Klopp’s side set a new top-tier record with 20 wins and one draw from 21 games to start a season, something no side in any of Europe’s top leagues has managed before.
Firmino arrowed a shot into the net eight minutes before the break and, while both teams had clear-cut chances in the second period, it was to prove the only goal of the game.
Tottenham remain eighth, now nine points off a top-four place, while Liverpool took full advantage of Leicester’s loss at home to Southampton to yet again stretch their lead at the top, and are now seemingly hurtling towards a first title win in 30 years.
Jose Mourinho handed a Premier League debut to 20-year-old defender Japhet Tanganga, with Danny Rose and Davinson Sanchez also restored to the starting side. In the absence of skipper Harry Kane, Heung-min Son and Lucas Moura led the line for Spurs.
Klopp recalled Alex Oxlade-Chamberlain for James Milner, the sole change from the side that had started against Sheffield United in the league nine days ago.
The visiting Reds almost got off to a blistering start. Spurs won a corner inside 60 seconds but they were so very nearly rueful as Liverpool countered at breakneck speed and Firmino’s shot was blocked, only for the rebound to find Oxlade-Chamberlain, who cannoned a shot off the post.
At the other end, Moura flashed a shot wide on five minutes in what was a bright opening to the contest.
From there it was the league leaders who asserted control on proceedings, with Mourinho’s shape typically set up to try and frustrate.
Midway through the opening half, Pablo Gazzaniga had to be alert to stop a header from Virgil van Dijk after a clever cross from Jordan Henderson.
Sadio Mane bounced a shot over the bar from inside the box soon after but it was Firmino who made the breakthrough.
Tottenham will feel aggrieved that a throw-in call went against them on the left-hand side but, on the restart, Firmino showed his class.
Salah’s toe-poked pass found the Brazilian and he opened up space eight yards from goal with a deft touch of his left foot before hammering the ball across Gazzaniga and into the net.
Salah could have doubled the lead before the break, while in stoppage time Moura shot over the crossbar as Tottenham ended the half as they began it, with a corner.
Serge Aurier and Son both had efforts off target as Tottenham enjoyed a rare period of dominance with the hour approaching, as Klopp sent on Adam Lallana in place of Oxlade-Chamberlain.
Mourinho summoned Erik Lamela and Giovani Lo Celso in place of Danny Rose and Christian Eriksen with 20 minutes to play – the Danish playmaker exiting to a very mixed response from the home support after another ineffective display.
Son flashed a shot over the crossbar with less than 15 minutes remaining as Spurs chased a leveller, while Lo Celso fired a low cross across the six-yard box but there was no one there to finish.
The Argentine then somehow managed to angle a shot wide off the outside of his boot as he stretched to meet Aurier’s teasing cross at the back post – much to Mourinho’s dismay.
They were two big moments for the hosts in quick succession and they were to rue their failure to convert as Liverpool almost doubled their lead late on via the dancing feet of substitute Divock Origi, foiled this time by Gazzaniga.
Klopp’s men eased through three minutes of stoppage time, with Xherdan Shaqiri coming on for Salah, and are now unbeaten in 38 Premier League games.