Spurs had to cope with a BVB onslaught in the first half as the Bundesliga leaders attempted to fight back after their 3-0 loss in the first leg in London.
Dortmund came up against a Hugo Lloris in inspired form however, with the Spurs keeper producing some exceptional saves to deny Marco Reus, Julian Weigl and England international Jadon Sancho.
Spurs' only real chance of the first 45 minutes fell to Son Hueng-min but he fired just wide on 31 minutes.
Dortmund may have been encouraged by their first half display despite their failure to break the deadlock but any chance of a comeback was ended just after the restart when Kane put Spurs ahead.
Moussa Sissoko picked out the striker after BVB failed to clear, Kane took one touch and then expertly curled the ball around Roman Burki in the Dortmund goal and into the right corner.
Kane's effort took the fight out of the German outfit and they failed to register another shot on target after the opener.
Spurs now go into the draw for the quarter-finals, which will take place on March 15.