The striker matched his haul against the Canaries at White Hart Lane from Boxing Day, where we also won by the same scoreline, to make it 15 goals in his last 18 matches with Dele Alli slotting home goal number seven of his Premier League season to hand us the perfect start inside two minutes.
It was all about the first period for us as Alli's quickfire opener was followed up by Kane's penalty on the half-hour. We had 14 efforts at goal to Norwich's three, struck the woodwork twice and could have been completely out of sight.
The second half showed a different side to the team as we played a containing game. Norwich came out with more energy but the closest they came to reducing arrears was when Sebastien Bassong's goal was - correctly - ruled out for off-side.
We looked more threatening on the break as the game progressed, Kane hit the woodwork again and it was no surprise to see the striker wrap it up for 3-0 in the final minute.
There were two changes from our last Premier League outing at Crystal Palace with Kevin Wimmer making his first league start in place of the injured Jan Vertonghen, while Kyle Walker came in for Kieran Trippier as Mauricio Pochettino rotated at right-back.
It was the perfect start for us as we broke the deadlock with our first attack. Norwich made hard work of clearing a corner and when Christian Eriksen fired a low shot in, Declan Rudd could only parry into the path of Alli who tapped in from six yards to stun the home crowd.
The goal meant Alli took the crown of the most goals scored for Spurs by a teenager in one season, a remarkable achievement on the first anniversary of his move from MK Dons.
The home side didn’t roll over but we had another glorious chance eight minutes later when Toby Alderweireld met an Eriksen corner with his head but planted his effort narrowly wide.
The home side looked nervous at the back but still posed a threat going forward and Dieumerci Mbokani stung the palms of Hugo Lloris with 15 minutes played.
Kane had a spring in his step and two minutes later he beat his man on the left and fired a cross in for Heung-Min Son, who clipped the outside of the post with a deft flick.
With 30 minutes gone we doubled our lead. Kane did well on the edge of the box to get a low shot in that Rudd parried into the path of the lurking Alli, who was bundled over by Bassong. The referee booked the former Spurs man and pointed to the spot, Kane duly sending Rudd the wrong way with the penalty to make it 2-0.
Ivo Pinto had the ball in the net soon afterwards with a clever header over Lloris, but the flag went up early for off-side.
It was all-action as Kane smashed an effort against the bottom of the post straight afterwards following more excellent work on the left.
We could have been out of sight with five minutes of the half remaining as first Alli squandered a header from a brilliant Danny Rose cross and then Eriksen fizzed an effort straight at Rudd.
There was one change at the interval with Nacer Chadli replacing Alli for us.
The home side started the second half better than the first with more of the game being played in our half, the lively-looking Steven Naismith going close with a header early on.
Kane had given Ivo Pinto a torrid time throughout and two minutes later he picked his pocket before charging towards goal. However, he dragged his shot wide from a good area.
Bassong thought he had pulled one back soon afterwards when he turned in from close range following a scramble in the box after a set-play, but again the flag was up for off-side.
It was becoming more of a dogfight which suited the home side, who peppered our box with high balls and pushed us back as we battled to weather the storm.
We didn’t retain the ball as well as we had done in the first half which made it nervy at times, although Tom Carroll – on as a substitute for Son – did bring something of a calming presence to the frantic midfield area.
We almost found a third goal eight minutes from time when Kane robbed Youssouf Mulumbu on half-way and surged through on goal but, having beaten Rudd, he saw his low shot rebound off the far post.
Mulumbu volleyed over for Norwich shortly afterwards but we finally completed the scoring in the 90th minute when Eriksen robbed Martin Olsson of possession, poking the ball through for Kane to run on and finish clinically.
Norwich (4-4-1-1): Rudd; Ivo Pinto, Klose, Bassong, Olsson; Howson, Tettey (c), Mulumbu, Jarvis (Jerome 70); Naismith (Odjidja 74); Dieumerci Mbokani. Substitutes (not used): Ruddy, Martin, Hoolahan, Redmond, Bennett.
Spurs (4-2-3-1) Lloris (c); Walker, Alderweireld, Wimmer, Rose; Dembele, Dier; Alli (Chadli 46), Eriksen (Lamela 90+1), Son (Carroll 70); Kane. Substitutes (not used): Vorm, Davies, Trippier, Bentaleb.
Goals: Spurs – Alli 2, Kane 30 (pen), 90.
Yellow cards: Norwich – Bassong 29, Tettey 89. Spurs – Alderweireld 58, Rose 88.
Referee: Kevin Friend.