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 scored from the spot in the 40th minute to put Spurs on track for a first league win.