The bottom-of-the-table Canaries twice led through Mario Vrancic and a Serge Aurier own goal as they aimed to breath new life into their hopes of Premier League survival.
Christian Eriksen’s brilliant free-kick and Kane’s spot-kick ensured Spurs did not slump to a third defeat in seven games, but they failed to take the chance to leapfrog Chelsea into the top four.
The area of the field with work to do for Jose Mourinho remains obvious and Spurs have kept just one clean sheet in 10 games since he took over and were guilty of some calamitous defending.
Norwich find themselves marooned at the foot of the table, six points from safety, but have played plenty of good football only to be found wanting in both boxes in their first season back in the top flight.
The Canaries slackness at the back could have cost them early on as Kane teed up Dele Alli to fire wastefully over before Kane was played in by one simple long ball by Toby Alderweireld, but Tim Krul raced from his goal to save.
At the other end, Vrancic could not believe his luck as Alderweireld stood off and allowed him to close in on goal before firing low past Paulo Gazzaniga to open the scoring 18 minutes in.
Only VAR prevented Tottenham falling further behind before the break as Jan Vertonghen allowed Teemu Pukki to control Vrancic’s ball over the top and the Finn finished clinically when one-on-one with Gazzaniga.
However, Pukki’s joy was short-lived as he was harshly adjudged to be offside after a VAR review.
Mourinho showed his displeasure with Spurs’ first 45 minutes by making two half-time substitutions and changing his system as Lucas Moura and Davinson Sanchez replaced Juan Foyth and Vertonghen.
The Portuguese got the upturn in performance he desired as only a brilliant block by Christoph Zimmermann denied Alli an equaliser just two minutes after the restart.
Eight minutes later, the visitors were level as Jamal Lewis handled on the edge of his own area to prevent Kane a clear sight of goal.
Eriksen has been a peripheral figure both under Mourinho and Mauricio Pochettino earlier in the season due to his contractual standoff with the club that will most likely see the Dane leave on a free transfer in the summer.
But he produced a timely reminder of his quality by curling the resulting free-kick into the top corner.
Moments later, the ball was in the Norwich net again but this time the offside flag went in the hosts’ favour as Alli’s classy finish was ruled out.
Just as Tottenham seemed to have taken control, the tide turned again in the other direction as Norwich retook the lead in comical fashion.
Aurier turned the ball into his own net after the ball ricocheted into his path from Alderweireld’s tackle on Pukki.
Spurs laid seige towards the Norwich goal in the final quarter and not for the first time this season Daniel Farke’s men were unable to see the game out.
Kane was wiped out by Zimmermann as he cut inside and the England captain picked himself up to dispatch the resulting penalty.