Madrid were leading 2-0 at 90 minutes, courtesy of first-half goals from Gonzalo Garcia and Fran Garcia.
Los Blancos were then given another scare as Dean Huijsen saw red for a foul in the box on Serhou Guirassy, who netted the subsequent penalty.
He ensured Madrid claimed victory and a place in the semi-finals, where they will face Paris Saint-Germain.
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‘Nobody told him’ – Jude Bellingham on missing ‘dream’ match-up against Jobe
Video credit: SNTV
Dortmund coach Nico Kovac added: “It is a great big save. That’s the reason why he is the best goalkeeper in the world.
“To be honest, it’s a deserved win from Real Madrid. We showed all our qualities and now it’s not a failure if you go home after defeat against Real Madrid.”
Courtois, meanwhile, said: “In football it’s like this. All the teams that are here, they compete until the end.
“A game could become like this at any minute and that’s what happened at the end.”
Next up for Madrid are PSG. It’s a game that will see Mbappe take on his former club for the first time since completing a move to Los Blancos last summer.
