Kane's two goals, the first from the penalty spot, means he now has 27 league goals in 2015 and has broken Teddy Sheringham's previous club record of 26.
Substitute Tom Carroll rounded off the victory with a superb second-half strike as Mauricio Pochettino's side move three points clear of Manchester United in the race for the Champions League
Norwich sit 17th.
It took four seconds for the first 'one our own chants' to break out around White Hart Lane as Kane continues to make light of those early one-season wonder predictions, his tally now 11 goals in the league and 13 in all competitions.
Dele Alli was the provider for both and the midfielder played another impressive supporting role before going off to a standing ovation late on.
Norwich, meanwhile, were unable to follow up their surprise win at Old Trafford and they now have 24 hours to prepare for a crunch basement battle at home to Aston Villa on Monday.
The Canaries were the brighter side early on as Vadis Odjidja-Ofoe's bending shot from distance was pushed wide by Hugo Lloris and then Cameron Jerome climbed highest at a corner but headed over.
Alli, however, was finding more and more space behind the Norwich midfield and it was his deft reverse pass that led to the opening goal, Kane nicking the ball around Rudd before feeling the clip of the keeper's gloves and tumbling over
There was no further punishment for Rudd given Kane's touch had taken him away from goal but referee Mike Jones pointed to the spot and the Spurs striker confidently drilled the penalty home.
The goal seemed to drain the belief out of Norwich as the half ended with Alli jinking past Russell Martin with a nutmeg and Toby Alderweireld going close with two free headers, the first saved well by Rudd.
Tottenham had thrown away 11 points from winning positions before kick-off, the second most in the league, so it was perhaps no surprise that Pochettino was raging after the restart, geeing up the crowd and hounding his players to go in search of a third.
Alli should have scored it after he played a sharp one-two with Erik Lamela but with only Rudd to beat, he found the keeper's legs.
The midfielder was involved again soon after as his driving run forward saw the ball fall kindly for Lamela, who tried an audacious rabona only for Rudd to dive across and make the save.
Spurs were rampant but Norwich hung on and the Canaries enjoyed a better spell after the hour.
There were cries for a penalty when Jerome was wiped out by Lloris in the area but the French keeper punched the ball away first and Jones appeared right to wave play on.
Any hopes of a Norwich comeback, however, were extinguished in the 74th minute when Carroll, three minutes after coming on as a substitute, smashed a swerving shot from 25 yards into the bottom corner.
As the 23-year-old, an academy graduate, celebrated his first league goal in a Tottenham shirt, the White Hart Lane faithful ended the match as they began, breaking into another rendition of 'one of our own'.
TWEET OF THE MATCH
@RodneyMarsh10: " This season Harry Kane has scored more goals than Depay, Martial & Rooney combined! £100m player !'" Harry Kane was among the goals again at White Hart Lane
https://twitter.com/RodneyMarsh10/status/680788054022467585
PLAYER RATINGS
TOTTENHAM
Hugo Lloris: 6
Kyle Walker: 7
Toby Alderweireld : 6
Jan Vertonghen: 6
Ben Davies: 6
Eric Dier: 6
Mousa Dembele: 7
Christian Eriksen: 7
Dele Alli: 9
Erik Lamela: 8
Harry Kane: 9
Substitutes
Tom Carroll: 7
Son Heung-min: 5
Nacer Chadli: 5
NORWICH
Declan Rudd: 5
Russell Martin: 5
Sebastien Bassong: 6
Ryan Bennett: 6
Martin Olsson: 5
Youssouf Mulumbu: 5
Alex Tettey: 6
Nathan Redmond: 6
Vadis Odjidja-Ofoe: 7
Robbie Brady: 6
Cameron Jerome: 7
Substitutes
Graham Dorrans: 5
Jonathan Howson: 5
Dieumerci Mbokani: 5
STAR PLAYER
Harry Kane: The striker's double means he now has 13 goals in all competitions for Spurs this season and also saw him break Teddy Sheringham's club record of 26 league goals in a calendar year
Another stirring performance from the striker.
MOMENT OF THE MATCH
Erik Lamela scored a stunning 'rabona' goal in the Europa League last season and he almost pulled it off again in the second half, shooting audaciously with his left foot only for Declan Rudd to make a diving save.
VIEW FROM THE BENCH
Mauricio Pochettino will be delighted with another impressive performance from his players, which makes it two consecutive wins since their shock home defeat to Newcastle a fortnight ago
Alex Neil's side struggled to cope at White Hart Lane but he will know Monday's clash at home to Aston Villa is his side's crunch game of the Christmas period.
MOAN OF THE MATCH
Kane should have had a hat-trick in the second half when Tottenham had four against one in the penalty area but they somehow managed not to score.
WHO'S UP NEXT
Watford v Tottenham (Barclays Premier League, Monday, December 28)
Norwich v Aston Villa (Barclays Premier League, Monday, December 28)
Source : PA
Source: PA