Tottenham were given a huge scare by Villa, who led for the bulk of the game at the Tottenham Hotspur Stadium through John McGinn's early strike.
But relentless pressure from the Champions League finalists eventually told as Ndombele equalised in the 73rd minute.
England captain Kane broke the visitors' hearts with goals in the 86th and 90th minutes.
Brighton's new manager Graham Potter enjoyed a dream start to his time in charge as the Seagulls beat last season's FA Cup finalists Watford 3-0 at Vicarage Road.
Abdoulaye Doucoure's own goal gave the visitors a half-time lead before Florin Andone and Neil Maupay on his debut sealed a highly impressive three points for a Brighton side that narrowly escaped relegation last season.
Burnley were also 3-0 winners at Turf Moor as an Ashley Barnes double and Johann Gudmundsson's strike condemned Southampton to a heavy defeat.
Crystal Palace's wantaway star Wilfried Zaha put his grievances with the club's hierarchy behind him to make an appearance as a second-half substitute against Everton, but neither side could break the deadlock in a 0-0 draw.
Everton were one of the biggest spenders in the window with seven new signings, but were forced to hang on for a point when Morgan Schneiderlin was sent off.
Speaking about Zaha's performance, Palace coach Roy Hodgson said: "He did well, set us up for a chance that could have won us the game.
"He's a little bit rusty, when a player comes back from three weeks away he's going to end a bit of time to find his true form. He's phenomenal when it comes to fitness."
Sheffield United grabbed a late point away to Bournemouth on their return to the top flight.
Chris Mepham put the Cherries in front just after the hour mark on the south coast, but Billy Sharp's strike two minutes from time salvaged a 1-1 draw for the Blades.
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