Wimbledon's Jimmy Abdou hit the crossbar with a curling first-half effort and Spurs needed 63 nervy minutes at Wembley before finally breaking down their League One opponents.
"Wimbledon played so well".
With Manchester City out of sight at the top of the Premier League and a fearsome Juventus to face in the Champions League next month, Tottenham can ill-afford any complacency this weekend.
Kane beat both Lionel Messi and Cristiano Ronaldo to claim the highest goalscoring tally of past year - 56 in total - and has attracted the attention of Zinedine Zidane, who will look to replace Karim Benzema at the end of the season.
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Wimbledon exhausted after that, with Vertonghen adding the third goal from range after robbing Lyle Taylor deep in opposition territory as Spurs saw out the closing stages comfortably. However, Kane may be given the day off, opening the door for either Fernando Llorente (two goals and one assist) or Son Heung-min (10 goals and three assists), with both forwards adept at leading the line for the London side.
Ziege said: "Obviously, he's scoring goals for fun".
Before that, the pattern of the game had quickly been set, with Spurs camped in Wimbledon's half as they attempted to break the visitors down.
Jan Vertonghen scored his first club goal in five years after Harry Kane had bagged a brace in the space of a couple of minutes.
Kane has established himself as one of the world's best players and finished 2017 with 56 goals for club and country which is more than any other player in Europe, including the likes of Lionel Messi and Cristiano Ronaldo.