Such was their superiority, Liverpool should have won by three or four goals and there was a late wobble when substitute Erik Lamela netted in stoppage time. But it will be a source of comfort to Klopp that his players can become better, even more clinical.
The visitors may count themselves unfortunate they were not awarded the lead inside the first minute, as Firmino flicked in James Milner's cross, but Sadio Mane had been in an offside position. Referee Michael Oliver ruled out the goal on the advice of his assistant.
A stray Eric Dier pass let in Mohamed Salah but Michel Vorm saved the forward's shot and the Dutchman prevented Sadio Mane from scoring after a loose ball from Mousa Dembele had gifted Liverpool possession.
But Vorm was at fault as Liverpool took the lead six minutes before halftime. Tottenham's back-up keeper misjudged his punch from Milner's corner and Dier only headed out as far as Wijnaldum, who looped his header back over Vorm and the ball crossed the line before it was clawed out of the net.
Wijnaldum could have doubled the lead moments later but dragged a low shot wide.
It was the move that resulted in Andy Robertson's cross dropping on to the top of the bar two minutes into the second half that had Klopp applauding his team, and Liverpool suffered their biggest let-off moments later before effectively wrapping up another three points. Lucas Moura, Tottenham's most dangerous player, brilliantly took down a Christian Eriksen pass and cut in past Joe Gomez but the Brazilian's low shot clipped the outside of the near post.
Four minutes after Moura's miss, it was game over for Mauricio Pochettino's team, as a high ball caught out Kieran Trippier, Mane got away down the left and his cross was turned against a post by defender Jan Vertonghen. However, Firmino was quick to react and turned the ball in.
It was at the end of September last year when champions Manchester City proved how serious they were by winning at Chelsea, and this felt like a similarly significant moment for Liverpool.
There were an unnecessarily nervy couple of minutes for Liverpool in stoppage time as Lamela pulled a goal back and fellow substitute Son Heung-min fired wide. Son appealed that he had been fouled by Mane just before missing the target but, other than the final 90 seconds, Klopp's players fully deserved his applause.
This was Tottenham's second defeat in a row since beating Manchester United 3-0, ahead of a Champions League trip to Inter Milan, and there remain worrying signs. That his team again conceded a goal from a set play will have angered Pochettino and Harry Kane did little to dispel the theory that he is running on empty.
"It's a collective problem and I won't point the finger at one player," said Pochettino.
"Look, I told you the challenge is massive this season. Yes, we need to give time to the players to get the best and to show our potential as a team. This game showed again that we need to improve if we want to be contenders to win some titles."
