The England striker made sure there was no repeat of last season's home woes in the competition with his side's second and third goals after Dortmund's Andriy Yarmolenko had cancelled out Son Heung-min's opener in a frantic start.
Kane had failed to find the net in August before scoring twice for England against Malta and twice at Everton at the weekend and he showed he is back to his razor-sharp best with a fine left-foot finish in the 15th minute.
He then slotted his second on the hour mark to double his Champions League tally to four and, more importantly, seal only Tottenham's third win in 13 matches at Wembley where two defeats in the competition last season proved so costly.
The only blemish on the night for Spurs was centre back Jan Vertonghen's dismissal in stoppage time.
"Harry Kane was fantastic," Tottenham manager Mauricio Pochettino told reporters. "It was so important to start well in a tough game.
"Dortmund played really well, they dominated the first half. We increased our level in the second half and we were very clinical in front of goal."
It was not a night for latecomers at Wembley with three goals in a breathless opening 15 minutes.
Tottenham, keen to start making the national stadium feel like home, if only for a season, went ahead after four minutes when Son burst down the left and with keeper Roman Burki perhaps expecting a cross the South Korean flashed home a left-foot shot that crept in at the near post.
It momentarily silenced the yellow mass of Dortmund fans parked behind the goal but they were in full voice in the 11th minute when experienced Ukrainian Yarmolenko, signed last month from Dynamo Kiev, equalised in stunning fashion.
After a neat one-two with former Manchester United midfielder Shinji Kagawa he looked up and curled an unstoppable shot into the top corner of Hugo Lloris's goal.
Tottenham were temporarily rattled but did not have long to fret as they restored their lead four minutes later with a virtual carbon copy of their opener.
This time Kane shrugged off the weak challenge of Omer Toprak and advanced into the area before lashing a left-foot shot across Burki who again probably could have done better.
Dortmund, who enjoyed 63 percent of possession, remained dangerous and Christian Pulisic was close to finishing off a counter-attack but could not divert Pierre-Emerick Aubameyang's cross goalwards from close range.
Aubameyang was unlucky to have his half-volleyed effort disallowed for offside early in the second half. "It was 100 percent a goal," Dutch coach Peter Bosz said.
Tottenham finally got the two-goal cushion they needed on the hour when Christian Eriksen slid in Kane on the left edge of the area and the striker's left-foot shot took a slight deflection before nestling in the far corner.
Home keeper Hugo Lloris then saved well from Aubameyang while Kane almost grabbed his hat-trick.
Tottenham played the final few minutes with 10 men after Vertonghen was shown a second yellow card for catching substitute Mario Goetze in the face with his arm.
He will miss Tottenham's next game against APOEL Nicosia.