The visitors were awarded a soft penalty inside six minutes as Dele Alli went down under the challenge of Kevin Long. Kane stepped up to the spot and calmly converted for the first of his goals.
It was not the first time Alli had irked the Burnley fans. In the fourth minute, referee Michael Oliver had yellow carded him for a fierce challenge on Charlie Taylor.
It was always going to be a tough ask for the Clarets to find their way back, but the task was almost made twice as daunting in the 17th minute, when Kane rounded Nick Pope only to fire into the side-netting.
Pope then made an alert double save to deny Moussa Sissoko five minutes later after Tottenham carved Burnley apart on the counter.
The hosts settled into the game as the first half progressed, but went into the break trailing nevertheless.
Tottenham started the second half off on the front foot and Burnley survived an early scare within the first two minutes. A cross from Christian Eriksen bounced off Pope and into Kane, deflecting inches wide of the post.
The North Londoners should have doubled their lead 55 minutes in, having hit Burnley with a well-orchestrated break, but Son Heung-min spectacularly fired over from close range after being teed up by Serge Aurier.
However, in the 69th minute, Spurs made amends. Sissoko reacted quickly after winning the ball to spot Kane and feed him. The striker coolly finished once again.
10 minutes afterwards, Kane scored his best goal of the night. After having won the ball, he raced forward to receive the decisive pass from Alli before netting for the third time.
Sam Vokes put the ball in the back of the net moments later only to see his header ruled out for a foul on Hugo Lloris.
Burnley may feel that was one of many 50/50 decisions which went against them on the night, but nobody can argue that Tottenham didn’t thoroughly deserve their victory. As a result of it, they have leapfrogged Sean Dyche’s side and moved into fifth place.
