Spurs made the 12-time European champions fight back for a 1-1 draw at the Bernabeu on Tuesday.
The Premier League side led when Kane's clever run forced Madrid defender Raphael Varane to divert Serge Aurier's right-wing cross into his own net.
The Spanish champions equalised before half-time when Aurier's rash challenge on Toni Kroos was punished with a penalty, Cristiano Ronaldo drilling in to extend his tally as the leading scorer in Champions League history.
"A point at the Bernabeu, you would take that every day of the week. It shows how far we have come as a team," Kane, 24, said.
"We are happy. Of course they are going to have chances but we played well and made it tough for them."
The home side increased their urgency immediately after the break as Spurs were pinned back, with visiting keeper Hugo Lloris producing stunning acrobatic saves to deny Karim Benzema and Ronaldo.
However, Spurs grew back into the contest and Kane - who has been linked with a move to the Bernabeu this week - almost scored a winner for the Premier League side.
The England striker found himself clean through on goal with 20 minutes left, but was denied by a superb fingertip save from keeper Keylor Navas.
The result moved Tottenham ahead of Madrid at the top of Group H at the halfway stage, having now scored more away goals than their opponents who also have seven points.
Significantly, the English side's hopes of reaching the last 16 for only the second time were strengthened by Apoel Nicosia's 1-1 home draw against Borussia Dortmund.
Tottenham are six points clear of 2013 finalists Dortmund and Cypriot side Apoel, who have both already lost to Spurs and remain winless.
(BBC)
