Mesut Ozil’s third goal in as many matches gave the Gunners an 11th-minute lead but Tottenham’s pressure paid off when Kane fired home a 56th-minute equaliser for his first goal in a north London derby.
The striker, under the watchful eye of England manager Roy Hodgson, then sealed victory for his boyhood club four minutes from time with a looping header into the far corner.
Mauricio Pochettino’s side leapfrog their fiercest rivals and are level on points with fourth-placed Manchester United, who play on Sunday.
The hosts had started strongly, having plenty of possession and coming close when Kane’s curling right-footed effort was tipped around the post by David Ospina at full stretch.
But Arsenal broke the deadlock when Danny Welbeck burst past Danny Rose on the right to pull the ball back to Olivier Giroud, whose mis-hit shot fell for Ozil to smartly volley home from inside the area.
Ospina was called into action again to tip away Rose’s low strike towards the bottom corner, with the full-back dragging a similar effort inches wide of the post soon after.
The Gunners ‘keeper needed to be alert to push Ryan Mason’s powerful strike away, while Nabil Bentaleb and Kane both saw efforts go wide of the post.
Spurs continued to dominate possession in the second half and eventually found a deserved equaliser when Ospina could only parry a flick-on from Christian Eriksen’s corner, Kane quickest to react at the far post to slide the ball home.
The hosts were on top, with Moussa Dembele and Bentaleb both having low strikes parried away by Ospina.
At the other end, Welbeck saw a bending effort tipped away from Hugo Lloris, with Laurent Koscielny heading straight at the Spurs ‘keeper when unmarked at the resulting corner.
But with the game heading for a draw, Kane planted a looping header from Bentaleb’s deep cross into the far corner for his 14th Premier League goal of the season, Spurs’ third league win in a row their most satisfying.
Tottenham: Lloris (7), Walker (6), Dier (7), Vertonghen (7), Rose (6), Bentaleb (7), Mason (8), Eriksen (7), Dembele (7) Lamela (7), Kane (9)
Subs Used: Chadli (6), Stambouli, Paulinho
Arsenal: Ospina (7), Bellerin (6), Mertesacker (7), Koscielny (6), Monreal (6), Coquelin (7), Ramsey (6), Welbeck (6), Cazorla (6), Ozil (7), Giroud (7)
Subs Used: Rosicky (6), Walcott (6), Akpom
Man of the match: Harry Kane