Kompany, who was booked during the game and will miss their last-four first leg against holders Paris Saint-Germain, said the Bavarians, who are chasing a treble of titles, had to dig deep to overcome record 15-time European champions Real 6-4 on aggregate in a classic tie.
"The boys were mentally strong today to recover from setbacks," Kompany said. "The fans helped us as well. We stayed calm and always felt that our moment would come. The boys deserved this win.
"What remains for me is the total togetherness that we showed. That's what we take from the game. We showed absolute belief and will to fight our way back into the game."
"During the game our team has a lot of talent and experience to deal with these situations. After 35 seconds in this game to concede a goal, you then have to press the reset button. We stayed focused on our game," Kompany said.
Hosts Bayern trailed 1-0 after 35 seconds when Arda Guler took advantage of a bad mistake by goalkeeper Manuel Neuer, and Real twice more took the lead to go into half-time 3-2 ahead.
Bayern, 2-1 winners of the first leg in Spain, kept their composure, however, and struck twice in the dying stages through Luis Diaz and Michael Olise to snatch victory and advance to the last four.
The six-time European champions can potentially clinch the Bundesliga title this weekend, with a 12-point advantage and five matches remaining, before facing Bayer Leverkusen in the German Cup semi-finals.
"It was a very emotional game," said Kompany, in his second season at Bayern. "We had a lot of possession and always had the feeling we could score. But Real Madrid remain Real Madrid. They have very quick players and are always a threat."
Kompany became the 11th coach to lead Bayern to the European Cup semi-finals.