Georginio Wijnaldum scored away from home in England's top flight for the first time and Roberto Firmino gave Jurgen Klopp's men breathing space in the 54th minute.
Mauricio Pochettino's Spurs were again out of sorts after their 2-1 loss to Watford before the international break, with substitute Erik Lamela's stoppage-time goal giving the scoreline a very flattering look for the hosts.
Liverpool thought it had the lead inside 45 seconds when Firmino touched James Milner's excellent cross beyond Michel Vorm, but Sadio Mane prodded at the Brazilian's effort from an offside position.
Vorm stood firm when a slack pass from Eric Dier allowed Mohamed Salah a run at goal but Tottenham's goalkeeper — playing once more in place of injured captain Hugo Lloris — was culpable when Liverpool took the lead six minutes before half-time.
After a botched punch from a corner, Vorm could only paw Wijnaldum's header away from behind his goal line.
More sloppy play from Tottenham after the break gave Vorm the opportunity to atone for his error with a sharp stop from Mane, before Lucas Moura found space at the other end to shoot against the outside of Alisson's right post.
A post was also involved in Liverpool's second as Jan Vertonghen slid in to block Mane's low cross, Vorm was unable to gather the rebound and Firmino converted from point-blank range.
It should have been 3-0 after the hour as Mane delayed his pass to Naby Keita and Vorm pushed the resulting shot over.
Firmino limped off before Vorm's mixed outing concluded with a stop from Salah.
Such wasteful finishing set up late drama as Lamela pounced on slack set-piece defending before Mane escaped a penalty shout for an untidy challenge on Son Heung-min.
A mouth-watering UEFA Champions League clash at home to Paris Saint-Germain awaits Liverpool before it welcomes Southampton to Anfield next weekend in the Premier League.
Spurs' forthcoming double-header consists of trips to Inter Milan and Brighton and Hove Albion.