The result moves Spurs onto six points after their first two Champions League group games and continues Kane's blistering form in the Premier League and Europe, having also scored braces against Everton, West Ham United and Borussia Dortmund already this month.
He has certainly established himself as one of the most devastating marksmen in world football and, although he has some way to go before reaching the stratospheric scoring accomplishments of Cristiano Ronaldo and Lionel Messi, he is making it increasingly difficult not to mention him in the same breath as the illustrious duo.
While Kane flopped in front of goal in August, he has found the net with ease in September. Here, we take a look at his stunning transformation from frustrated frontman to the Premier League's most in-form forward.
Consistency the key
Yes, Kane struggled in August. But his impressive productivity stretches back far beyond the start of this season. In fact, his scoring exploits since the turn of the year have been truly exceptional. With 34 goals in all competitions in 2017, he has scored more goals than Southampton, Swansea, Burnley, Watford, Stoke City, West Bromwich Albion and Crystal Palace. He is, quite simply, a magnificently consistent goalscorer.
Last season, Kane finished the season by scoring four times against Leicester City before clinching a hat-trick against Hull City on the final day to reach 29 Premier League goals, beating Romelu Lukaku to the Golden Boot.
The England international also managed to improve on his 2015/16 tally of 25 goals, while his 2014/15 haul was 21. That shows a clear and steady progression for Kane who, under the sagacious tutelage of Mauricio Pochettino, has developed into perhaps the most feared striker in England.
Masterfully grasping his positional sense
While he started the season slowly, his recent form has shown Kane at his very best, a striker brimming with confidence and self-belief. Having those and the technique to fire the ball beyond the goalkeeper are essential ingredients to any striker's toolkit, but Kane's understanding of where to be in the box – and when to be there – has afforded him the chance to score in the first place.
But don't be fooled: Kane popping up in the right places is strictly by design. Under Pochettino, the striker has embraced the art of leading the line by working hard on the timing and shape of his runs, reading the game, learning when to pass and zoning in on the perceived weaknesses in the defenders he's facing.
Kane is blessed with a natural gift for finishing, but a bolstered understanding of the tactical side of the game in recent years has helped take his game to dizzying new heights, wherein he is being compared to Wayne Rooney at his peak.
And, yes, his positional sense has been key. Kane's second goal in the 3-0 win at Everton earlier this month illustrates how he has mastered the bread-and-butter move of a striker. Picking up the ball, he looked up, spread it to Ben Davies on the left and immediately endeavoured to dart into the box. However, Kane's intelligence as a striker enabled him to run into exactly the right position.
With Real Madrid transfer target Dele Alli at the far post, Kane picked up the perfect spot six yards out, in between a napping Morgan Schneiderlin and Ashley Williams, who was reluctant to leave Alli in space.
Arriving just as Davies curled an excellent cross in, Kane opened up his right foot and finished with aplomb. While his first goal in that game – an attempted cross that just happened to curl into the top-corner – was sheer luck, this was a goal borne of thought, composure, and killer instinct.
Using space to his advantage
That knowledge of space was demonstrated in the win over Dortmund, too, as Kane's double lifted Pochettino's men to an impressive win over the Bundesliga outfit.
With the game delicately balanced at 2-1, Kane struck his second of the game to crush Dortmund's hopes and wrap up the points.
And that goal was another telling example of Kane's impeccable movement in the final third. Christian Eriksen receives the ball, which is usually Kane's queue to make a move. However, with Davies taking it upon himself to surge through the middle, the striker took up his position on the left, with Davies occupying the attention of two Dortmund defenders. Eriksen finds Kane, who drives a shot unerringly into the corner.
Both of Kane's goals in the tense 3-2 win against West Ham United were products of predatory instinct, too.
The opener, an expertly-judged glancing header from Alli's cross, came after Kane took advantage of Angelo Ogbonna not dropping off, running off the defender before surging into the box and giving himself a free header, with which Joe Hart had no hope.
Four minutes later, Kane scored again, this time making sure he was in exactly the right place to benefit from Hart saving a one-on-one with Alli. Kane, having escaped the attention of Winston Reid, calmly slotted the ball home to double Spurs' lead.
Kane's deceptive power and pace
When Sky Sports recently put a graphic comparing Kane to Sergio Agüero, Lukaku and Álvaro Morata, it was surprising to see that the Spurs man actually had a faster top speed than Agüero, who is far from a slouch.
While speed isn't an attribute you would immediately associate with the Spurs man, he has shown how he can use his physical capabilities to devastating effect this season.
His first goal against Dortmund was a thrilling snapshot of his ferocious power and pace, which makes him a nightmare to come up against if you are a defender.
On the halfway line, Kane out-muscled one Dortmund player, before surging past another, shifting the ball to the left and unleashing an unstoppable shot into the roof of the net for a masterpiece of a goal.
In seeing off three Dortmund players single-handedly, Kane proved that, while Alli and Eriksen are good to have around, sometimes he simply doesn't need any support.
Three brilliant goals. Three important points. What a night for Spurs! ?
Kane ⚽️ pic.twitter.com/6t5mtmkQBi
— Football on BT Sport (@btsportfootball) September 13, 2017
Composed, focused, decisive
The most difficult thing for a striker to master is maintaining composure in front of goal. Thierry Henry once said that he tried to do the same thing every time he had the chance to score (a trademark side-footed slot into the corner), and while Kane has certainly shown a variety of finishes, they all have the same characteristics running through them: composure, focus and a decisive, ruthless edge that leaves keepers in despair.
It's no mistake the vast majority of Kane's goals are perfectly-placed. On Tuesday night, he thrilled with a glorious three-peat. His first goal saw him rolling the defender before dispatching the ball into the far corner. His second goal was one touch to set and a second to find the same corner. His third was a glancing header which nestled into, you guessed it, that far corner.
? Left foot
? Right foot
Harry Kane with the perfect Champions League hat-trick.
Class ??? pic.twitter.com/IKfUJ0gnMz
— Football on BT Sport (@btsportfootball) September 26, 2017
It was a perfect hat-trick (left foot, right foot, header) but it was also a beautifully simple treble, dispatching the ball into the same place with a minimum of fuss. It also encapsulated Kane as that rare striker who is comfortable on both feet and deadly in the air.
As the weeks go on and we continue to see Kane wheeling away in celebration, clutching match balls and winning points for Spurs, it's safe to conclude that he is operating at a level unmatched by most of his peers.
While it's easy to roll your eyes at fantastical comparisons to Ronaldo and Messi, the stats simply support Kane's place in football's highest talent bracket.
For instance, Kane has a better shots-to-goals ratio than both of the superstars in the calendar year of 2017. Also, Kane's career total of 82 goals in 119 Premier League appearances looks favourable compared to Ronaldo's 84 in 196.
He really is looking like the complete striker, and what an asset he will prove to be for Pochettino this season. It's also important to remember that he is still only 24.
Historically, strikers realise their full potential somewhere between 26-30, so with that, it could be expected that Kane will only continue to improve, should he stay injury-free. Considering what we have seen in recent weeks, the prospect of an even better Kane is exhilarating if you are a Spurs fan, and frightening if you are not.