The Spanish giants produced a 3-0 victory over the Germans at home last week and that three-goal advantage proved vital as they were able to progress to the semi-finals with a 3-2 aggregate.
Marco Reus was almost the hero for Dortmund, scoring a first-half double, before providing Henrikh Mkhitaryan with a great chance to tie things up only to see the Armenian miss from point blank range.
There was a let-off for BVB in the 16th minute when Angel di Maria's penalty was well-saved by Roman Weidenfeller and the home side made the most of that when Reus pulled one back from the three-goal deficit in the 24th minute.
The German midfielder pounced on Pepe's poor header towards Iker Casillas, nipping in to slot home after rounding the Madrid keeper.
13 minutes later and Dortmund were just a goal down when Reus was on hand to fire home after Robert Lewandowski's initial effort had cannoned back off the woodwork.
There were plenty of other chances for Dortmund, particularly in the first half when Mkhitaryan missed the first of two sitters for the night, but Carlo Ancelotti's Los Blancos were able to see it through to book their spot in the elite four.
Juergen Klopp's side were brilliant in the return leg, against a Madrid side missing Cristiano Ronaldo, but it was not enough as the nine-time champions continue on in their pursuit of an elusive 10th European title.
2013/14 UEFA Champions League - quarter-final second leg
*Real Madrid wins 3-2 on aggregate
Signal Iduna Park, Dortmund, Germany - 65,829
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