After a sharp piece of work on the edge of the hosts' box by Giovani Lo Celso, Son Heung-min's angled finish put Jose Mourinho's men ahead after 27 minutes.
Eleven minutes after half-time, Matt Ritchie's screamer flew into the top corner from a tight angle to level things up.
But Kane restored Spurs' lead only four minutes later, heading in from Steven Bergwijn's fine cross.
Kane headed in again in the 90th minute to put his side out of sight, finishing at point-blank range after Erik Lamela's shot was saved.
Newcastle remain 13th on 43 points. Spurs climb to seventh with 55 points, one behind Wolverhampton Wanderers.
Newcastle: 20 Jul v Brighton (A)
Spurs: 19 Jul v Leicester (H)