The England international opened the scoring after seven minutes when Dele Alli drew a foul from Kevin Long inside the hosts' penalty area, winning a spot kick. Kane duly dispatched from 12 yards, sending Clarets keeper Nick Pope the wrong way.
Spurs looked comfortable from there on out but wasted several gilt-edged chances before Kane doubled up the win in the 70th minute. Moussa Sissoko won the ball on halfway and sent a defence-splitting pass through for Kane to tuck away on the run.
Kane then made and finished his third, winning the ball deep in Burnley's half and exchanging passes before firing low past Pope into the bottom corner.
If he scores in Tottenham's Boxing Day fixture against Southampton he will beat Shearer's record of 36 goals in a calendar year.
Pope deserves England call
Gareth Southgate was in the crowd at Turf Moor with the assumption he was casting his eye over Tottenham's clutch of England players. Really, though, the Three Lions boss could do worse than sitting up and paying attention to the form of Clarets stopper Pope.
A Premier League rookie, the 25-year-old was playing in the Isthmian Premier Division as recently as 2012. This term he has been a revelation between the sticks for Dyche's side in the absence of Tom Heaton who has been sidelined since September with a dislocated shoulder. In that time Pope has become one of the Premier League's most underrated keepers, conceding just three times before today.
Given Joe Hart‘s abysmal form and subsequent benching at West Ham United, Fraser Forster‘s struggles at Southampton and Heaton's injury woes, it is baffling that Pope has not been worth a spot in Southgate's squad before now. That must change when the next round of international fixtures arrive in March.
The former Charlton Athletic youngster was at his best again against Spurs, twice denying Moussa Sissoko in the first half with a brilliant double save which underlined his razor-sharp reactions once more.
Dele Alli pantomime villain once more
The headline last weekend was Manchester City's dominance as the champions-elect brushed Spurs aside 4-1 to continue their march towards the title. The crushing nature of the win ensured little coverage was given to poor challenges by Alli and Kane, both of which could have resulted in red.
This time there was little getting away from it. It took Alli just four minutes to fly in on Long at Turf Moor, escaping with a booking for a challenge that might have merited red on another day.
Soon after the natives were even more infuriated when Alli won the penalty for the visitors' opener by going down under the slightest jarring of knees inside the box.
Alli must be careful. For all his undoubted talent he is rapidly building himself a reputation as trouble. That is a reputation which is easier to gain than lose and will, subconsciously or otherwise, inform the view of referees and make him a target for opposing players.
This is, after all, hardly the first time we've seen this side of his game.
Tottenham's full-back roulette
Kyle Walker‘s £55million move from Spurs to Manchester City left Tottenham light at right-back. So much so that youngster Kyle Walker-Peters had to start at No.2 on the opening day of the season, catching the eye at Newcastle.
Inevitably, though, there was a clamour for Pochettino to go out and spend some of that money on a first-team ready replacement for Walker. Eventually, he did just that.
But the signing of Serge Aurier from Paris Saint-Germain appears to have boxed the Argentine coach into an awkward corner where he feels obligated to give the Ivorian starts his performances have not justified.
Kieran Trippier made the right-back or right wing-back spot his own last season. He cannot have expected to find himself in and out of the side this term after seeing off Walker in the second half of last season. The last time he started consecutive Premier League fixtures was either side of the September international break.
Burnley fail to deal with Spurs press
As the game wore on a clear pattern emerged. Tottenham's high press forced Burnley backwards and when their centre-halves were in possession the Clarets were nervous, losing the ball on several occasions.
After an hour when Kane pinched the ball from Long deep in the home side's half and slid the ball through for Christian Eriksen. But the pass was ahead of the Dane who could not punish Burnley.
It was not the first time Burnley had been caught out and if Spurs were to build on their lead it looked their most likely route to another goal. That proved to be the case with Kane's hat-trick goal, a move started by Spurs' No.10 winning the ball just outside Burnley's penalty area.
The flip side of Spurs' press meant Burnley struggled to get any service into striker Chris Wood because so many attacks were cut off early. The Kiwi forward saw next to nothing of the ball before being withdrawn with injury after 35 minutes. His replacement, Ashley Barnes, fared little better.
Sánchez reminds Spurs what they missed
Suspended for last weekend's thrashing at Manchester City, Davinson Sánchez's presence at centre-back was clearly missed as the leaders breezed to a 4-1 victory.
Centre-back was probably not a position Spurs fans would have prioritised strengthening over the summer but the club record £42million Spurs paid to sign Sánchez from Ajax has proved a bargain.
Although Burnley offered very little going forward at Turf Moor, the Colombian was back to his imperious best immediately, dealing with the physical threat of Wood, Barnes and Sam Vokes. Whenever there was a half chance he was on hand to snuff it out. How they could have done with him a week ago.