Bayern won 4-3 on the night - and the quarterfinal 6-4 on aggregate. Real had Eduardo Camavinga sent off on 86 minutes with the visitors leading 3-2. Arda Guler, who scored twice on the night, also saw red after the final whistle for his protests.
"Today we showed a lot of character; we had a lot of faith to rise above adversity,” Kompany said afterwards.
“We can leave with our heads held high."
Regarding whether PSG will be an even tougher opponent in the semis: “That’s difficult to answer… they and Real Madrid are the biggest right now; it’s something you can’t really say."
Kompany continued: "A picture is worth a thousand words. And I'll always remember that final celebration with the fans. That image. Today we showed a lot of character. We had a lot of faith to rise above adversity.
"Real Madrid is very, very strong; they have enormous individual quality. It's a very special night for us."
Kompany was booked on the night and will be suspended for the first-leg of the semifinal.
He also said: "I'm not happy with my yellow card. There were a lot of emotions involved, and the fourth official should listen to us, too. When there are arguments, I understand that if bad language is used, a caution should be given, but that wasn't the case.
"And some things happened on the pitch that weren't right. I said it yesterday, and I stand by it: the three-match ban is too harsh. Look at all the players Real Madrid had on yellow cards today, too. My yellow card was very strict. But oh well, it's not important. I have a lot of faith in my staff , in the whole club."