Last season's Championship play-off winners threatened a huge upset for a long time, leading through John McGinn's strike on nine minutes.
Dean Smith's side frustrated their hosts for more than hour before Tottenham's record signing Tanguy Ndombele curled home a leveller with 17 minutes remaining.
Kane – so often a slow starter in past seasons – then notched a quick-fire double within four minutes of full-time to ensure Mauricio Pochettino's team eventually won with something to spare.
Villa hit the front with a goal of real simplicity. Tyrone Mings launched a long punt forward and McGinn darted across Danny Rose to claim it. The Scotland midfielder controlled the ball and turned into space inside the penalty box before slotting past Hugo Lloris.
McGinn was again involved as Trezeguet threatened to double the advantage midway through the half, while Spurs had their chances to level just before half-time when Kane headed over from six yards following good work from Ndombele.
Moussa Sissoko passed up a glaring chance to restore parity shortly after half-time and Villa were starting to believe it might be their day when Tom Heaton brilliantly parried away a goal-bound header from Davinson Sanchez midway through the second half.
They failed to clear their lines properly, however, and Lucas Moura was able to tee up Ndombele to curl home from 20-yards – ironically as Pochettino appeared set to hook his record signing.
Heaton was again at his best to thwart a free-kick from Christian Eriksen, but the stopper was powerless when a poor through pass cannoned off Bjorn Engels into Kane's path for the Spurs skipper to coolly arrow the ball into the corner.
As the game entered stoppage time, Sissoko's pass found Kane on the edge of the box and he set himself before curling a low shot beyond Heaton into the far corner to ensure Spurs' day ended well.