Bayern's narrow win, thanks to a 63rd-minute header from Joshua Kimmich, gave Bayern a 3-2 aggregate win after a 2-2 draw in London in the first leg.
The hosts worked hard for the win which was in stark contrast to the free-flowing draw at the Emirates.
England captain Kane told TNT Sports: "It's an unbelievable win - it's been a tight season for us.
"We had to fight and grind and today was that. We knew we could make the difference with our own fans.
"It was what we expected, a tough game, a tight game and we got a great goal from Josh to make the difference."
Bayern's form in Europe is in stark contrast to that in the Bundesliga, where they are 16 points behind Xabi Alonso's Bayer Leverkusen, who last week won the title, breaking the Bavarians' 11-year streak in the process.
"Our expectation is to try and win the Champions League," added Kane.
"It's going to be a tough game in the semi whoever we play.
"There was maybe some tempo missing in the Bundesliga. That can't happen next season."
