Erik Lamela did grab a stoppage-time consolation for Tottenham, who were made to rue the absence of captain Hugo Lloris with stand-in goalkeeper Michel Vorm arguably culpable for both goals in a second straight defeat.
Tottenham's defensive setup played into Liverpool's hands and they were fortunate not to go behind within 45 seconds when Firmino's effort was chalked off because Sadio Mane was offside and adjudged to be interfering with play.
Spurs gifted the Reds a handful of chances, but Mohamed Salah could not capitalise on a slack pass from Eric Dier as he shot straight at Vorm from 12 yards. More sloppiness from Mousa Dembele released Mane who was held up by Toby Alderweireld and could not generate enough power to trouble Vorm on his left foot.
The Dutchman made some decent saves but his weakness under the high ball was exploited on 39 minutes when he made a mess of punching clear James Milner's corner. Wijnaldum looped the second ball over Vorm who flapped at it again, but the header had already crossed the line.
It was Wijnaldum's 19th Premier League goal but first away from home in 58 matches on the road.
Lucas Moura was Tottenham's only attacking threat and he nearly single-handedly dragged them level shortly after the restart when he cut in from the left flank, dribbled past three defenders and shot against the post from 15 yards out.
That could have been Tottenham's route back into the match but three minutes later they were two down. Mane peeled away from Trippier and squared the ball into the six-yard box, where Jan Vertonghen diverted the cross onto the post, and when Vorm failed to gather the rebound Firmino had the simplest of tap-ins.
With Mauricio Pochettino operating a midfield diamond and pushing his full-backs high up the pitch, there was plenty of space for Salah and Mane to exploit but some selfish decision making prevented Liverpool from opening up more of a cushion.
That gave Spurs some hope to have a go in the final 10 minutes, with Lamela shooting wide from the edge of the box, and Harry Kane getting a header on target which was easy pickings for Alisson, who had a quieter day than he may have expected.
Lamela set up a nervy finish with his 93rd-minute reply, controlling a corner on his chest at the back post and volleying through Alisson Becker from a tight angle.
Son Heung-min had half a chance to steal Spurs a point but took too long on the ball and failed to get a shot away. The South Korean was caught by Mane in going to ground and could have had a last-gasp penalty in a repeat of the 2-2 draw at Anfield last season.
This is the first time since 1990-91 that Liverpool have won the first five league games of the season, and the first time in over two years that Tottenham have lost back-to-back Premier League matches.
SPURS (4-4-1-1): Vorm; Trippier, Alderweireld, Vertonghen, Rose; Dier (Wanyama, 83'), Dembele (Lamela, 60'), Winks (Son, 73'), Eriksen; Moura, Kane
LIVERPOOL (4-3-3): Alisson; Alexander-Arnold (Matip, 90'), Gomez, Van Dijk, Robertson; Wijnaldum, Milner, Keita (Sturridge, 83'); Mane, Salah, Firmino (Henderson, 74')