Tottenham's Christian Eriksen nudged the ball into an empty net on the rebound to give the visitors an early lead after a superb piece of hold up play from Son Heung-min allowed Erik Lamela to fire at goalkeeper Bernd Leno.
Arsenal had most of the ball but Spurs were far more threatening on the break and Leno was forced to keep out stinging efforts from Kane and Eriksen before Tottenham were awarded a penalty for a clumsy foul by Granit Xhaka on Son.
Kane came close to sealing the points for Tottenham when he struck the post but Arsenal became galvanised when Spanish midfielder Dani Ceballos came off the bench and smashed a shot which keeper Hugo Lloris did well to tip over.
Arsenal's Sokratis Papastathopoulos had a goal ruled out for offside while Tottenham's Moussa Sissoko spurned a last-gasp chance in a thrilling finale by blasting over the bar. Arsenal have seven points while Spurs are on five from four games.