A second of the evening for Kane, and a third for Tottenham, arrived on the hour through a well-taken strike just moments after the visitors wrongly had a goal ruled out.
Relive how the 90 minutes of action unfolded with Sports Mole's live text coverage below.
STARTING XI: Lloris; Alderweireld, Sanchez, Vertonghen; Aurier, Dier, Dembele, Davies; Eriksen, Son; Kane
SUBS: Vorm, Foyth, Trippier, Walker-Peters, Winks, Sissoko, Llorente
STARTING XI: Burki; Piszczek, Sokratis, Toprak, Toljan; Sahin, Dahoud, Kagawa; Yarmolenko, Aubameyang, Pulisic
SUBS: Weidenfeller, Subotic, Castro, Gotze, Zagadou, Isak, Philipp
Mauricio Pochettino: "I think Wednesday is so important, to start well in the competition and get the three points will be key to the future. In theory it should be Real Madrid first and the others trying for second, but football is not about theory. Our ambition is to win every game. Real Madrid are on a different level: they won the last two Champions Leagues so it's a different reality. I think it'll be a very competitive group."
Peter Bosz: "For the way in which we play we need to train regularly. I believe that my players understand better what I want. But it could also be the case that we concede three or four goals from time to time with our attacking style. My players are playing well and they have confidence going into the Champions League game against Tottenham. They also have confidence in the way we play and that is very important."
TOTTENHAM HOTSPUR SUBS: Vorm, Foyth, Trippier, Walker-Peters, Winks, Sissoko, Llorente
BORUSSIA DORTMUND SUBS: Weidenfeller, Subotic, Castro, Gotze, Zagadou, Isak, Philipp