The Bavarians looked beaten when the French struck twice in two minutes early in the second half to open up a 5-2 lead.
Bayern, however, bounced back with goals from Dayot Upamecano and Luis Diaz in a three-minute span to make it 5-4, considerably improving their chances of turning the tie around next week in Munich.
Atletico Madrid play Arsenal in the other semi-final on Wednesday.
"It'll be who takes their moments next week," said Kane, who put the visitors in front with a 17th-minute penalty. "There were a lot of moments out there today, and it's probably going to be the same next week. So, us being at home with the crowd behind us, we hope that can push us over the line."
"Overall, we had moments where we could have killed the game earlier on as well. So, we take a lot of pride in the fact that we got back to 5-4 because away from home, being 5-2 down could be a really tough place to be. But we fought and we're back in the tie."
Kane, who became the first English player to net in six successive Champions League fixtures, now has 13 goals in the competition this season.
The match will go down as an instant classic after becoming the highest-scoring semi-final leg in Champions League history.
"We had moments where we could have been more clinical in the final pass or the final finish," Kane said. "We felt we got better and better, especially as the game went on, they started to tire. We've just got to bring the same intensity. We're going to need more of that next week."
"I think we saw two high-level teams out there today, especially in the attacking play and in the transitions, the speed, the intensity in the one-against-one battles. Two of the top teams going toe-to-toe," said the England captain.