It wasn't as easy a game as the scoreline suggests as the League One side were well in the game for just over an hour but two goals in two minutes from Kane ended their resistence in the game.
Jan Vertonghen then added a third goal six minutes later with a deflected long-range strike which means that Spurs will now take their place in the fourth round draw which takes play on Monday evening.
Pochettino made plenty of changes to his side but it was still very strong
As expected Mauricio Pochettino made plenty of changes to his starting eleven for the game with Kyle Walker-Peters and Juan Foyth handed a rare start by the Argentinean, while Kane captained his side at Wembley.
Spurs frustrated by their plucky visitors throughout the first half
In terms of the game itself, Spurs started the game on the front foot but found it difficult to break through a compact Dons defence as they had to result in taking shots from long range.
The Lilywhites had upwards of 80% possession throughout the half but only tested George Long on a couple of occasions. Erik Lamela was at the heart of both of these chances as firstly he saw his volley well-saved by Long after Fernando Llorente chested the ball into his path.
Lamela's other chance came from a lightening counter attack which seen Kane give the ball to winger, who tried to chip Long but the keeper wasn't having any of it as he made the save.
The Don's went within inches of taking a half-time lead at Wembley
The best chance of the first half though actually fell the way of the visitors as a brilliant curling effort from Nadjim Abdou was brilliantly saved by Michel Vorm, who tipped the ball onto the crossbar.
The visitors grew in confidence after that without having much of the ball but at half-time, they would have been delighted with how things went for them given the fact that Spurs would have been confident of scoring some early goals in the game.
Vorm was called into action once again at the start of the second half
The Dons continued to look bright at the start of the second half as a Jonathan Meades header from a free-kick was heading into the back of the net but for a very good save from Vorm once again.
That chance though seemed to wake the hosts up and they came within inches of taking the lead when Moussa Dembele hit a long range strike towards goal but with the keeper beaten, the ball came back off the post.
Kane struck twice in two minutes to settle any nerves for the hosts
Everyone in the ground though could sense that the goal was coming though for Spurs and it did finally arrive in the 63rd minute when Kane fired the ball home from close range after Moussa Sissoko passed the ball to him.
The visitors barely had time to digest the first goal whenever they conceded again two minutes later and it was that man Kane once again as he fired home past the helpless Long.
Vertonghen's first goal since 2015 sealed the win for Spurs to send them through to round four
It got even better for Spurs six minutes after the second goal when Vertonghen hit a fantastic long-range strike towards and via a deflection the ball flew into the top corner of the net to seal the cup tie for the hosts.
In their search of getting more goals before the end, the only real chance that the hosts created fell the way off Heung-Min Son, who saw his effort from just inside the box come back off the post with the keeper beaten.
Therefore, the game ended 3-0 to Spurs but the League One side should take some credit from the game as they really made Spurs work hard to open the scoring before they went on to score a couple more in quick succession.