Vinicius Jr scored a hat-trick, with Antonio Rudiger and Lucas Vazquez also netting. All Real's goals came in the second-half after Donyell Malen and Jamie Gittens had BVB 2-0 ahead at the break.
Vini Jr was named man-of-the-match and spoke to the press afterwards: "We believe in ourselves, we know that at home with our fans anything can happen.
"We arrived at the changing rooms very quiet, we listened to the manager (Carlo Ancelotti) and we said one thing, if we score the first we will come back again. Thanks to the fans but we have to improve and play like this from the start because if not the manager won't hold out.
"The manager always finds a way to make the games easier for us. We changed the formation and also the dynamic in the second half. It's our competition and we want to win it again."
He also said: "I can always do more, I'm 24 years old and I want to stay here forever and give a lot to this team that has given me everything."
On the Ballon d'Or chants, he added: "I'm happy when the fans sing my name, it's a dream come true and I can score goals for them."