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.
"We came up against a superior team, there is no doubt about that," Villa coach Dean Smith said. "They were Champions League finalists for a reason. We knew it would be a tough baptism."
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.
"When Tanguy got his goal it was full throttle to the end," Kane said. "We created a few chances and thankfully a couple came to me and I was able to put them away." For sports news , updates, live scores and cricket fixtures , log on to indiatoday.in/sports . Like us on Facebook or follow us on Twitter for Sports news, scores and updates. Get real-time alerts and all the news on your phone with the all-new India Today app. Download from Do You Like This Story? Awesome!