The back-to-back Bundesliga winners now sit top of Group D after Marcel Schmelzer scored the winner to earn the Germans a 2-1 victory over the La Liga champions after Cristiano Ronaldo had cancelled out Robert Lewandowski's opener.
However, Subotic admits that his side won't have long to dwell on the win.
"What can I say? It's the kind of thing you dream of as a kid," said Subotic. "It was clear ahead of the game that we were up against one of the best teams in the world and we wanted to show we could play on a par, which we managed to do. To get a victory like this against a team like that in front of your own fans – it's something that will remain long in the memory.
"It was extremely difficult to stay focused against players of that quality. They can pick out the most intricate passes. You can't afford to give them a yard of space. We punished them for their mistakes and did well, but it was really, really hard work.
"If we keep playing like this we can go far, but I don't want to look that far ahead yet. For now, I'm just happy that if my career ends tomorrow, I can say to my kids that we beat Real Madrid. Unfortunately there'll be no partying tonight, though. It's straight to sleep for me."
The two sides meet in the reverse fixture in two weeks at the Santiago Bernabeu.