Tottenham began the night ahead on goal difference and will rue some golden chances after the break but Firmino's goal sparked joy amongst the 2,000 fans allowed in at Anfield.
Theo Walcott scored against his old club for Southampton who moved into third spot with a 1-1 draw at struggling Arsenal who, for the second match running, ended a game with 10 men.
Everton backed-up their weekend win over Chelsea to beat fourth-placed Leicester City 2-0 away and move into fifth spot.
Leeds United struck three times in the last 11 minutes to sink Newcastle United 5-2 at Elland Road.
Fulham edged out of the bottom three with a 0-0 draw at home to Brighton and Hove Albion.
All eyes were on Anfield though where Jose Mourinho's Tottenham were to face the biggest test yet of their title credentials against Juergen Klopp's champions.
A fascinating clash of styles ensued with Liverpool dominating possession but Tottenham carving out enough chances to have ended Liverpool's long unbeaten home run in the Premier League, which now extends to 66 matches.
Liverpool went ahead through Mohamed Salah's deflected effort but Tottenham levelled from their first attack when Son Heung-min raced on to Giovani Lo Celso's through ball and drilled a shot past Alisson.
The visitors paid dearly for their missed chances as Brazilian Firmino sent a bullet header past Hugo Lloris in the final minute of normal time.
Mourinho and Klopp had a lengthy discussion after the final whistle with the Spurs boss saying he told his Liverpool counterpart that the "better team did not win".
"I thought he was joking, but he wasn't," Klopp said.
Arsenal snapped a run of four successive home league defeats as Pierre-Emerick Aubameyang cancelled out Walcott's opener at the Emirates Stadium but the hosts are now six matches without a league win and pressure is growing on manager Mikel Arteta.
He was not helped by a red card for Gabriel after he hauled down Walcott in the 62nd minute.
"We showed some resilience and character to hang in there for so long against a really dangerous team," said Arteta.
Richarlison gave them a halftime lead and Mason Holgate doubled their advantage after the break.
"It's been a good week. We've found consistency defensively and that was the reason for these two results," Everton manager Carlo Ancelotti said.
"The season is long but at the moment it's good."
Leeds and Newcastle were locked at 2-2 after 77 minutes but goals from Stuart Dallas, Ezgjan Alioski and a cracker from Jack Harrison secured the points for Marcelo Bielsa's vibrant side.