The England striker gave the home side the lead from the penalty spot in the 26th minute at White Hart Lane, before making it 2-0 three minutes before half-time with his second goal of the game.
Tom Carroll then added gloss to the scoreline with a goal in the 80th minute to keep Spurs in fourth place in the Premier League table.
Mauricio Pochettino’s men do not have long to recover from the Boxing-Day clash, with the north London side due to travel to Watford in the Premier League on Monday afternoon.
And speaking to BBC Sport after Saturday’s win, Kane expressed his delight at helping the Lilywhites keep their bid for Champions League qualification on track.
Kane said: “[It was] a very good performance by the team. We had a disappointing result here last time against Newcastle but a solid result this time.
“We are delighted with the three goals and the win. Norwich are a side fighting for every point so we knew we had to come out in the second half and keep the tempo up. We managed to get the third goal and kill the game off.
“We go again on Monday away from home and need to pick up as many points as we can.”
Kane has now scored 27 Premier League goals in 2015, a new club record that eclipses the previous best from Teddy Sheringham.
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