It might have been different had Jonjo Shelvey's 14th-minute free-kick not struck the post and bounced clear.
Newcastle continued to threaten and Hugo Lloris was forced into a diving save to keep out a Jamaal Lascelles header.
But after Kane mishit a one-on-one with Martin Dubravka, the striker curled Son Heung-min's pass into the top corner to move to within three of Mohamed Salah in the race for the Golden Boot.
Newcastle stay 10th having already secured their PL status.
Spurs: 13 May v Leicester (H)
Newcastle: 13 May v Chelsea (H)