Sofiane Boufal did pull one back, but Kane completed another record - becoming the first player to score six Premier League hat-tricks in a calendar year - before Dusan Tadic's late consolation did little to lift Southampton's festive spirits, the visitors having now lost three of their last four top-flight games.
Shearer was quick to congratulate Kane, tweeting: "You've had a magnificent 2017 @HKane".
7 Got his seventh hat-trick of the year on Saturday. "That's what I always have done throughout my career and it is what I will continue to do".
Once decried as a one-season wonder, the 24-year-old English marksman has gradually evolved into the No.9 par excellence, thriving for both his club (Tottenham Hotspur) and country (England).
Putting his efforts into wider context; across Europe's top five leagues, only Lionel Messi (40) has found the back of the net on more occasions.
67 Games he took to reach 50 Premier League away goals.
Some Saints fan will have felt slightly hard done by considering their side had hit the woodwork on two separate occasions, and they managed to find some shred of a consolation for the travelling supporters.
93 The second-highest EPL scorer in Spurs' history, behind Teddy Sheringham (97). It's hard to say he's the best but he's shown to everyone he's one of the best.
His hard work was nearly thrown away immediately as Saints showed that they were far from out of the contest, good build-up play comes to Boufal whose effort can only be parried by Hugo Lloris.
Mousa Dembele and Danny Rose came in for Moussa Sissoko and Ben Davies while Pellegrino opted to replace the banned Charlie Austin with Shane Long, who remains without a goal since February. Burnley skipper Ben Mee's poor ball out of defence was pounced upon by Moussa Sissoko, who sent Kane through to slot beyond Nick Pope.
"It's an wonderful story for us, as a coaching staff, when you look at him play - not only feel happy [but] so proud of all this he has achieved", said Pochettino, who added that Tuesday match was so open it could have finished 7-4 to Spurs.
The retired star himself has backed him to break the mark against Southampton tomorrow. Scoring goals for fun.
"I've improved physically - recovery for games, getting ready for each game, eating right and ice baths", added Kane. "Now, it's more than double that".