Centre-back Dejan Lovren suffered a nightmare start which began after only four minutes when Kieran Trippier played the ball over his head, giving Kane time to run into the box, round Simon Mignolet and fire home past two defenders.
With Liverpool floundering, Lovren was culpable again as a long throw from keeper Hugo Lloris sailed over his head in the 12th minute, enabling Kane to set up Son Heung-Min for Spurs' second.
Son was within a whisker of adding a third soon afterwards, latching onto a glorious Christian Eriksen pass and crashing an angled shot past Mignolet but back off the bar.
The South Korea international then almost turned provider as Spurs continued to pour forward, breaking at pace but seeing his attempted pass to Kane cut out by Joel Matip.
With 23 minutes gone, and out of nowhere, Liverpool found a way back into a game that could already have been beyond them.
Kane appealed in vain for a foul when he was dispossessed near the halfway line and Henderson floated an outstanding pass through to Salah, who sent a bobbling finish past Lloris as the Spurs keeper opted to stay on his line.
Spurs had a chance to restore their two-goal lead as the half-hour approached, Son again in the thick of the action but seeing his weak strike blocked by Mignolet.
Lloris gathered a low cross from Salah with Liverpool players waiting to pounce as Jurgen Klopp's side began to apply some pressure, but Kane almost made them suffer again when a fine run ended with a skidding low shot wide.
With the first half into its third added minute, Liverpool cracked as Matip's poor attempt to head a cross clear fell to Dele Alli and he steered the third Spurs goal beyond Mignolet.
Oxlade-Chamberlain fired in a dangerous cross as Liverpool attempted to start the second half on the front foot, Salah just unable to get to it, but more chaos in the visiting defence was not far away as Spurs wrapped up the points before an hour had gone.
This time, Mignolet flapped at a free kick and although Jan Vertonghen's effort was blocked on the line by Roberto Firmino, it fell to Kane who took one touch to control before lashing home.
Only an extraordinary save from Lloris prevented Liverpool from pulling a goal back as the game approached its final 20 minutes, with the France international diving to tip Philippe Coutinho's fine strike from the edge of the box onto the bar.
Kane was substituted late on after going to ground and clutching his hamstring, with a trip to Manchester United next up for Spurs.