It was a result few had foreseen, particularly when Diego Costa put Jose Mourinho's men ahead midway through the first half.
But Kane, pushing for an England call up, leveled with an exquisite individual goal and Danny Rose put Spurs 2-1 ahead after he reacted quickly when Nacer Chadli's shot hit the post and rebounded back into play.
Kane was then fouled by Gary Cahill with Andros Townsend scoring from the penalty spot to give the home side a 3-1 lead at the break.
It was 4-1 shortly into the second half when Kane's neat footwork set him up for a fourth, but still the points were not sealed.
A mistake by Federico Fazio allowed Eden Hazard to play a one-two with Cesc Fabregas to reduce the deficit to two goals.
Tottenham nerves were finally eased when Kane played a neat pass to Chadli for the Belgian international to score the fifth only for John Terry to grab his third in four games for Chelsea to complete an eight-goal thriller.
Earlier, Manchester City joined Chelsea on 46 points with the same goal difference after a headed goal from former Chelsea star Frank Lampard gave the reigning title holder a 3-2 win over Sunderland.
Lampard, whose extended loan to City from New York City was confirmed on the eve of the match, grabbed a late winner after City had thrown away a two-goal lead at the Etihad.
The 36-year-old former England midfielder connected with Gael Clichy's cross to secure a vital three points with a seventh home league win of the season.
City looked to be cruising to victory after a superb strike from Yaya Toure and a second from Stevan Jovetic.
But Sunderland hit back with goals from former City players Jack Rodwell and Adam Johnson, who scored from the spot.
In the early kickoff on New Year's Day, Manchester United was held to its second straight draw of the festive period in a 1-1 draw at Stoke.
The home side went ahead after just two minutes through defender Ryan Shawcross but Radamel Falcao leveled for United in the 26th.
United, held goalless at Tottenham last weekend, remain third in the standings and are unbeaten in 10 games, but manager Louis Van Gaal was unhappy with the display.
"We didn't deserve any more than a draw," he moaned. "They were closer to a winning goal than us. We should have played better today."
Struggling West Bromwich Albion has also confirmed the appointment of former Stoke and Crystal Palace boss Tony Pulis as its head coach.
Pulis, widely praised for lifting Palace into mid-table last season before leaving before the start of this campaign, watched from the stand as his new team secured a fighting 1-1 draw at West Ham.
Southampton kept up its challenge for a Champions League spot by inflicting a damaging 2-0 defeat on Arsenal to stay fourth.
Two goalkeeping errors from Wojciech Szczesny allowed Sadio Mane and Dusan Tadic to score in either half for the Saints, who withstood late pressure from Arsenal to secure a vital three points.
In the relegation battle, bottom side Leicester secured an unlikely point after hitting back from two down at Liverpool to grab a 2-2 draw, while Burnley hit back three times to level 3-3 at Newcastle.
It denied Newcastle caretaker boss John Carver three points after taking over from Alan Pardew, who is set to link up with Crystal Palace and was at Villa Park to watch them battle to a goalless draw.