Dortmund have reason to be upset, though, as they were on top for large parts and had a goal wrongly ruled out for offside with the score at 2-1.
A second victory in 12 Wembley outings lifts the North London outfit joint-top of the Group H standings alongside Real Madrid, who were victorious at home to APOEL, but the win was soured somewhat by an unfortunate red shown to Jan Vertonghen late on.
Tottenham got the ball rolling with a goal from the first shot of the evening, with Son getting off the mark for the season by beating Roman Burki at his front post from a tight angle.
The visitors responded brilliantly, throwing plenty of bodies forward and getting their reward seven minutes later through a piece of individual magic from Yarmolenko, who marked his debut with an edge-of-the-box curler past Hugo Lloris.
Four minutes is all Spurs had to wait until they restored their lead, as once again Burki was somehow beaten at his front post by the power of Kane's angled drive after the Englishman ran past a couple of opposition players.
The final ball was proving to be BVB's downfall, with former Tottenham target Aubameyang putting too much on his right-sided cross when Christian Pulisic had peeled away from his man in the middle.
Pulisic thought that he had levelled up the match for a second time from the final meaningful kick of the half, only for the referee to blow his whistle for an offside infringement in the build-up.
Spurs saw two chances come and go in quick succession early in the second half, first through Kane who blasted over the bar from inside the box, before Son curled over when picked out in a good position.
It should have proved costly as Die Borussen, who won the only previous European tie between these two sides, found the net just short of the hour mark.
Aubameyang timed his run to perfection at the back post to volley the ball past Lloris, but the offside flag was wrongly raised to cut short his celebrations.
Moments later Kane had the ball in the net for a second time, well and truly putting his August misery to one side by tucking past Burki - his fourth Champions League goal.
Wing-back Ben Davies tested Burki as the Lilywhites closed in on another, while at the other end Lloris was required to keep out another Aubameyang volley down low.
Christian Eriksen and Kane both saw chances come and go to wrap things up in the final 15 minutes, the latter dragging wide via a small deflection, as Spurs saw things through for all three points despite the red awarded to Vertonghen in added time for his stray elbow on Mario Gotze.