Kane's status as the man-of-the-moment looked fully justified when he burst into the box to win a penalty six minutes into stoppage time at White Hart Lane and stepped up to convert on the rebound after his initial effort was parried.
As well as sending the home crowd into raptures, it completed a superb comeback for the hosts who had trailed 2-0 to goals from Cheikhou Kouyate and Diafra Sakho, before Danny Rose's scrappy effort heralded a late cavalry charge.
There was no need for last-gasp heroics for Liverpool, who took the lead after three minutes through Philippe Coutinho's stunning curler that caught the underside of the crossbar on its way past Southampton keeper Fraser Forster.
The match was dripping with controversy, however, as the hosts had three penalty appeals turned down in the driving south coast rain while Liverpool keeper Simon Mignolet handled outside the area and escaped a red card.
The three points that lifted Liverpool into sixth above Tottenham were wrapped up when Raheem Sterling converted from close range with 17 minutes remaining.
Liverpool are now a point clear of Tottenham on 45, one behind Southampton and two off fourth-placed Manchester United who occupy the final Champions League qualifying spot.
Bottom club Leicester City came painfully close to pulling off a surprise comeback win at Everton but were pegged back to draw 2-2 when Romelu Lukaku's header was deflected in by Matthew Upson with two minutes remaining.
Steven Naismith had given 12th-placed Everton a second-half lead before goals from Leicester's David Nugent and Esteban Cambiasso turned the match around but they could not hold on and are now four points adrift at the foot of the table.
