Kane scored in the first-leg of their 2-1 Champions League quarterfinal win. The England captain played despite carrying an ankle injury picked up on international duty.
Ahead of tonight's Bundesliga clash with St Pauli, Kompany said: "Harry Kane was fine after the game; he played a little beyond his pain threshold . But there was no reaction the next day, and that's the most important thing."
Kompany also discussed the future of veteran goalkeeper Manuel Neuer, who was outstanding at the Santiago Bernabeu.
He added, "I won't talk about his future out of respect for a top-level career and a top-level club. But I can talk about what he has already achieved with Bayern.
"When I arrived, I had (Thomas) Müller, Neuer, and (Joshua) Kimmich as captains, which made my job easier. Someone like Manu is more in the background, but he only has to say one sentence to bring order. That's why his role as a leader is just as important to me as it is on the pitch."