Kane scored one of the best goals of his 47 in that century, showing sheer strength and then clinical finishing to open the scoring on 40 minutes.
Three minutes later, Dele Alli finished from Kyle Walker’s smart cross and we were firmly in control, with the lads seeing out the second period with our clean sheet intact - thanks in part to an incredible stoppage time save from Hugo Lloris to deny James Ward-Prowse.
Mauricio Pochettino made two changes on his return to his former club as Mousa Dembele was preferred in midfield ahead of Tom Carroll while, in defence, Danny Rose missed the trip with illness meaning Ben Davies came in to replace him.
It was overcast and gusty on the South Coast yet surprisingly mild for the time of year as we entered a busy festive period.
The home side started on the front foot with some early pressure and it took the boot of Toby Alderweireld - back at the club at which he spent last season on loan - to block an early effort from Sadio Mane.
Mane looked a threat and forced a brilliant low save from Lloris five minutes later after Dusan Tadic did well on the left-hand side.
It was an early storm that needed to be weathered as the Saints piled on the pressure and made life difficult for us when we had the ball.
Step by step we started to get a foothold in the game, though, with some better play in the Saints’ half as Christian Eriksen flashed a dangerous ball across the face of goal with 15 minutes gone.
The home side remained a threat on the counter, however, and we had Jan Vertonghen to thank midway through the half when he scrambled down low to block another Mane effort as he spun inside the box.
Alderweireld went close at the other end with a header from a Walker cross that landed on the roof of the net, yet Paulo Gazzaniga had enjoyed a largely quiet spell in the Saints’ goal.
That was until 10 minutes before the break when he was called into action after Eriksen played an exquisite one-two with Kane before the latter fizzed a low shot in on the angle that stung the palms of the keeper.
Kane soon shattered the deadlock in style with a quite brilliant goal. Credit is due to Alli who stretched to poke a ball into the striker’s path – and took a nasty whack for his troubles. But after that it was all down to the quality of Kane. With three defenders in his path, he wriggled past the first two then poked it round Virgil van Dijk to take him clean through before opening up his body and slotting past the goalkeeper.
There was barely time to draw breath before it was 2-0. Alderweireld made a timely interception and fed Walker on the right. With plenty of options, he took his time and picked out Alli lurking at the back post, who took one touch to control the ball before rolling it home.
It was a whirlwind three minutes that silenced the home fans and meant we headed down the tunnel with a 2-0 lead at the interval, an outcome that had looked unlikely for most of the half.
The heavens opened for the second period with a torrential downpour as both sides remained unchanged.
We started the second period as we ended the first with plenty of possession and movement as we went in search of a third goal.
The home side sought a route back into the game with Graziano Pelle becoming more of a threat, although he failed to test Lloris with a couple of chances.
Substitute Tom Carroll might have extended our lead on 84 minutes but he couldn't get any power behind his effort and Gazzaniga made a routine stop.
We looked comfortable in the latter stages but there was one final action of note – and another save for the ever-growing Lloris scrapbook. Ryan Bertrand got away down the left and his cross found Ward-Prowse, who connected sweetly from 12 yards but the goalkeeper was ready and down to his left to palm away.
That was that - a fine pre-Christmas performance.
Southampton (4-2-3-1): Gazzaniga, Cedric, Van Dijk, Fonte (c), Bertrand; Clasie (Ward-Prowse 78), Wanyama; Mane, Davis (Long 64), Tadic (Juanmi 64); Pelle. Substitutes (not used): K Davis, Yoshida, Romeu, Targett.
Spurs (4-2-3-1): Lloris (c), Walker, Alderweireld, Vertonghen, Davies; Dembele, Dier; Lamela, Alli (Carroll 78), Eriksen (Chadli 86); Kane (Son 90). Substitutes (not used): Vorm, Wimmer, Bentaleb, Winks.
Goals: Spurs - Kane 40, Alli 43.
Yellow cards: Southampton - Fonte 40; Spurs – Alli 72.
Referee: Kevin Friend.