Luis Diaz and Kane struck to have Bayern 2-0 ahead in their first-leg quarterfinal tie before Kylian Mbappe pulled a goal back for the final score to finish 1-2.
Kane said afterwards: "We knew that coming to Madrid and trying to get a result is always difficult. We played some really good stuff and we could have done even better – maybe the final ball, the final finish, we had some good chances. But credit to Madrid as well."
Kane also praised goalkeeper Manuel Neuer, who was named man-of-the-match on the night.
"We can be really happy with a win," the Bayern top scorer added to TNT Sports. "We've enjoyed some really good moments together. We're at the business end of the season now, so we're hoping we can maintain that. We always feel dangerous, we always feel like we can score – and we did again away tonight."
Kane added: "We're in a good position but it's a one-goal lead and it can change really quick. We have to stay focused. We have a big Bundesliga game away (to St. Pauli) on (Saturday), so now it's just about recovering and focusing on that.
"The time will come next week to try and perform like we did today."