Mesut Ozil gave Arsene Wenger's side the advantage early on but the young striker's 21st and 22nd goals of the season turned things around after the break at White Hart Lane
Mesut Ozil had given the visitors the lead early in the first half with a brilliant finish to a Danny Welbeck cross flicked into his path by Olivier Giroud, though Spurs were rarely without threat to reply.
Such a response was finally delivered through Kane in the 56th minute when the young striker was the first to pick up on the rebound from David Ospina's smart save from a header at a corner.
And Kane was on call again in the dying minutes of normal time when he rose to meet Nabil Bentaleb's cross with a superb header past Ospina to secure the points.
It brought to an end Arsenal's five-match winning run in all competitions, Tottenham leapfrogging their rivals into the top four.
The hosts edged a cagey opening, with Ospina forced to tip wide Kane's early effort before the Colombian easily collected Danny Rose's disappointing delivery from the left.
However, in Arsenal's first purposeful foray forward, the recalled Welbeck burst past a surprisingly sluggish Rose to square for Giroud in the centre. The Frenchman cleverly helped the ball onto Ozil, with the Germany international able to angle his body and volley past Hugo Lloris to score for a third game running.
An increased intensity from both sides saw chances flow before the half-hour mark, with Rose threatening Ospina's goal twice more down the left. However, the Arsenal defence stood firm to soak up Spurs' pressure - wasteful finishing from Christian Eriksen and Ryan Mason helping their opponents' cause.
Welbeck was required to clear Eriksen's dangerous free kick late in the first half and Spurs started the brighter after the break.
A lack of cutting edge in the final third continued to prove their downfall until Kane drew his side level with a cool finish at the back post after Ospina had failed to palm Erik Lamela's corner out of danger.
The goal was no more than Spurs' pressure deserved but Welbeck almost put Arsenal back in front soon after - his curling effort kept out at full stretch by Lloris.
Ospina was called on again to deny Kane - having initially saved from Nabil Bentaleb - but the 21-year-old would not be denied as he further cemented his status as Spurs' latest fan favourite.
Kane's 22nd goal in all competitions came courtesy of a wonderful arching header from Bentaleb's cross as the striker impressed again in front of watching England manager Roy Hodgson.