Georginio Wijnaldum opened the scoring for Liverpool at Wembley before Firmino doubled the visitors' lead ten minutes after half-time.
A late goal from Erik Lamela proved to be just a consolation for Spurs, who were without captain Hugo Lloris and Dele Alli.
Liverpool finished 25 points behind City last season as Pep Guardiola's side became the first to ever break the 100-point barrier in the Premier League.
"It is only the job to do sometimes because we have players that work together that much longer".More news: World Cup champion goalie Hugo Lloris pleads guilty to drunken driving
"The players and staff onboard the club flight back to Liverpool sent the Brazilian a video wishing him well", the team said. It was a performance to back up manager Eddie Howe's pre-match assertion that his side, who have won three of their first five games, have closed the gap on the division's top teams.
"We should be talking in a different way now because if it was given, it's 2-2 if we score the goal".
Mane made a good run that caught Tottenham right-back Kieran Trippier out of position before sending over a low cross that Vertonghen diverted against his own post.
"Now we have to prove it and do it again, again and again".
"I'm really happy and proud about the performance because the boys came back (from worldwide duty) only on Wednesday and Thursday", Klopp said.More news: New Apple Watch Series 4 steals the show
"The performance was better than the result and the result is flawless", he said. "It was a really outstanding performance that the boys delivered today". We had 85 brilliant minutes, although unfortunately we didn't finish the game off.
"Actually I don't like to compare players but we had similar players and questions about Robbo (Andrew Robertson) and Alex Oxlade-Chamberlain", he said.
The Spurs chief continued: "First of all, I want to say they deserved their victory".
And Wijnaldum has warned his team-mates they need to improve if they want to be closer this time around.
Liverpool boss Jurgen Klopp had been concerned about Firmino when he spoke to the media immediately after the Spurs game.More news: Rockfall in Greece Leaves Seven People Injured
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