The defender claims BVB raised their game to beat the Liga champions, while goalscorer Schmelzer is emotional in victory, revealing that the match "fulfilled a childhood dream"
The famous win took the home side to the top of Group D, one point ahead of Los Blancos.
The Germany international said his team-mates showed a level of commitment against Jose Mourinho's side that had been strangely absent in their Bundesliga games of late.
"It's clear that against a team like Real Madrid we had to give everything," the 23-year-old told Sky.
"That's what we lacked in a couple of Bundesliga matches, when we thought we could play at half speed. It was a great game and a great result."
Fellow defender Marcel Schmelzer was the hero for Dortmund, netting the winning goal in the second half.
"Today has fulfilled a childhood dream, helping Borussia Dortmund to beat Real Madrid at home," the left-back said.
"To have scored the winning goal is something that I still cannot believe," he added.