Reece James seemed to have sealed the points for the hosts with a 77th-minute strike but Kane rose to head home a corner in the sixth minute of stoppage time.
As it happened: Chelsea vs Tottenham Hotspur
Chelsea manager Thomas Tuchel and his Spurs counterpart Antonio Conte, a former Chelsea boss, were both shown red cards after the final whistle as tempers boiled over on a sweltering afternoon in west London.
Centre back Kalidou Koulibaly marked his home debut in style by volleying Chelsea in front from a corner in the 19th minute and the hosts totally dominated the opening period.
Chelsea remained in the ascendancy in the second half and should have had the points wrapped up but were rocked when Pierre-Emile Hojbjerg equalised in the 68th minute.
Late drama at Stamford Bridge 🍿#CHETOT https://t.co/TpTyKMehDZ
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Tuchel was furious that the goal was allowed, believing Kai Havertz had been fouled in the build-up, and clashed with Conte.
Havertz then spurned a glorious chance to put Chelsea back in front but the hosts did reclaim their lead in the 77th minute when James fired past Hugo Lloris from a pass by Raheem Sterling — a goal that sent Tuchel joyfully racing down the touchline.
Tottenham, who began the season in optimistic fashion, could have had few complaints if they had gone home empty-handed but Kane’s late intervention means they have four points from two games, the same as Chelsea.
Tottenham have won only once in their last 38 visits to Stamford Bridge in all competitions.