Mesut Ozil???s third goal in as many matches gave the Gunners an 11th-minute lead, with David Ospina having to make a string of saves to keep the visitors in front.
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 four minutes from time with a looping header into the far corner.
The hosts started strongly, having plenty of possession and coming close when Kane???s curling right-footed effort was tipped around the post by Ospina at full stretch.
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 and Kane was quickest to react at the far post to slide the ball home.
The hosts were on top, with Moussa Dembele and Nabil Bentaleb both having efforts parried by Ospina.
At the other end, Welbeck saw a bending effort tipped away from Lloris, with Koscielny heading straight at the Spurs keeper when unmarked at the resulting corner.
With the game heading for a draw, Kane planted a looping header from Bentaleb???s deep cross in to the far corner for his 14th Premier League goal of the season and to give Spurs their third league win in a row.??