Following a 3-0 first leg drubbing at Wembley, Spurs travelled to Germany on Tuesday for the second leg of their last 16 tie with Dortmund.
The hosts pushed for a goal throughout a scoreless first half, but it was Kane who provided the breakthrough in the third minute of the second half.
Dortmund were caught napping at the back as Moussa Sissoko played in the England forward, who faced Roman Weidenfeller all alone.
Kane provided a simple finish inside the near post, effectively giving Spurs a five-goal lead with less than a half to play.
The forward scored twice against PSV in November to take his tally to 23 goals in Europe, and it was Tuesday's finish that saw him move into sole possession in Tottenham's goalscoring chart on 24 goals.
The November finishes saw him pass Martin Chivers on 22 and Tuesday's finish pushed him past former leader Jermain Defoe on 23.
Chivers is one of just three players ahead of Kane on Spurs' all-time list of finishes, with Bobby Smith and Jimmy Greaves the only other players with more goals for Tottenham in all competitions.
As well as his Champions League efforts, Kane has 16 goals and four assists in 25 Premier Leauge appearances for Spurs this season.
He also provided a goal in the Carabao Cup, taking his total to 22 goals in all competitions this campaign.
Following Tuesday's match, Spurs will turn focus towards Saturday's clash with Southampton in the Premier League.
Spurs currently sit third in the league, with Manchester United just three points behind in fourth place.