Mesut Ozil had given the visitors the lead early in the first half with a brilliant finish to a Danny Welbeck cross flicked into his path, though Spurs were rarely without threat to reply.
Such a response was finally delivered through Kane in the 56th minute when the young striker was the first to pick up on the rebound from David Ospina's smart save from a header at a corner.
And Kane was on call again in the dying minutes of normal time when he rose to meet Nabil Bentaleb's cross with a superb header past Ospina to secure the points.
