Erik Lamela received one for Tottenham in injury time, but it was too late to rescue an undeserved draw.
This mature exhibition was a true declaration of intent by Liverpool, who made his best start since 1990-91 while pursuing a first title in English from 1989-90.
It is not the first time this season that the Reds triumphed thanks to hard work, a superbly exercised defense and an opportunistic ending instead of repeating the kind of boisterous game that led them to the Champions League final last season.
But Klopp, winner for the first time at Wembley, will not complain after securing an away win in one of his six top rivals, proving that they are likely to be the biggest threat to the Manchester City champions.
For Tottenham, this was a disheartening second consecutive loss that served as a major blow to his own title ambitions.
Mauricio Pochettino had accused Tottenham of treating their 2-1 loss to Watford as a "testimony".
Challenging his players to show a killer instinct, Pochettino expected an angry response.
But, on the other hand, Tottenham almost stays behind after less than 60 seconds.
When James Milner hit a cross into the penalty area of Tottenham, both Firmino and Sadio Mane were able to run out of control thanks to Pochettino's sleepy defense.
Firmino, in an onside position, deftly threw his shot through Vorm and into the far corner, only to have Mane out of the game while trying to get the final touch.
The lethargy of Tottenham was not limited to his defenders and the English midfielder Eric Dier was lucky to get away with a terrible backpass that gave the opportunity to Mohamed Salah.
Salah ran through the goal, but he lingered too long, allowing Jan Vertonghen to press him into a pitch that was too close to Vorm.
It was the kind of golden opportunity that Salah would have buried during his campaign of 44 record goals last season.
But Tottenham was neglected from start to finish and they paid the price in the 39th minute.
Christian Eriksen's whimsical ringleader gave away a corner and, from Milner's side, Vorm, substitute for the injured Hugo Lloris, made a handful of attempts to clear.
Dier tried to avoid the danger, but his weak header only came to Wijnaldum and the Dutch midfielder nodded towards the goal, where Vorm had his wrong angles, allowing the ball to cross the line before he could get away.
Liverpool briefly disconnected when Lucas Moura almost conjured a draw with a low shot that shot off a post.
But in addition to cautioning his players for their defeat at Watford, Pochettino was forced to defend Harry Kane this week amid growing concern over the English striker's fatigued display.
Once again, Kane was a peripheral figure and, with Dele Alli lost through injury, Tottenham lacked teeth.
Liverpool sniffed the blood and doubled its lead in the 54th minute when Tottenham came down from behind.
Mane made a good run that stopped right-back Tottenham Kieran Trippier out of position before sending a low cross that Vertonghen deflected against his own post.
Vorm, mistaken for Vertonghen's intervention, was face down when the ball passed through the outstretched arms to Firmino, who could not fail on the goal line.
Lamela lashed home in the final moments, putting a flattering glow on a score that did not reflect Liverpool's dominance.