Arsene Wenger’s side broke the deadlock in the first half thanks to Aaron Ramsey’s close-range finish at White Hart Lane.
But Arsenal midfielder Francis Coquelin was sent off early in the second half after a second yellow card and Tottenham defender Toby Alderweireld levelled in the 60th minute.
England striker Kane put Tottenham 2-1 up with an outstanding finish from an acute angle to leave Spurs temporarily top of the table before Alexis Sanchez equalised with 16 minutes remaining.
Former Spurs striker Klinsmann, who scored a couple of spectacular goals during his Tottenham career, posted a Twitter message for Kane after his world-class finish against the Gunners in north London.
— Jürgen Klinsmann (@J_Klinsmann) March 5, 2016
Kane has scored 17 goals in 29 Premier League appearances for Mauricio Pochettino’s side so far this season.
The Tottenham striker has been linked with a move to Manchester United over the past 12 months.
Spurs are in action against Borussia Dortmund in the first leg of their Europa League last-16 clash on Thursday night.
MORE: The latest football news