Tanguy Ndombele also struck on his debut as Mauricio Pochettino's side dug deep to overcome a stubborn Villa team.
Kane, who missed a good chance in first-half injury time, capitalized on a defensive lapse from Jack Grealish before coolly slotting the ball past goalkeeper Tom Heaton in the 86th minute.
The England forward then grabbed his second in the 90th after being set up by Moussa Sissoko. It was just reward for Spurs' pressure after the home side chalked up 28 efforts on goal, compared to six from the visitors.
Earlier, John McGinn stunned the home crowd by scoring against the run of play in the ninth minute. He held off Danny Rose's challenge and shot low past goalkeeper Hugo Lloris following a long ball forward from Tyrone Mings.
Ndombele tried his luck from distance before Pochettino brought on Danish playmaker Christian Eriksen in the 64th.
Spurs kept attacking and club-record signing Ndombele finally leveled in the 73rd when he let fly through a host of bodies after Lucas Moura had laid the ball back.