With a 3-0 first-leg advantage, Madrid were rocking at half-time as Marco Reus' double reduced the aggregate score to 3-2 in a fervent atmosphere at Signal Iduna Park on Tuesday.
Both goals came after Angel di Maria had seen a penalty saved by Roman Weidenfeller, the Argentinian losing his footing on the wet surface as he went to strike the ball.
Two efforts from Reus - capitalising on mistakes from the visitors each time - gave Jurgen Klopp's side hope of a memorable fightback but they could not force the third goal that would have sent the tie into extra-time as Real gained a measure of revenge for last year's semi-final loss to Dortmund.
The visitors wasted a golden opportunity to practically wrap up the tie in the 17th minute, Weidenfeller diving to his left to keep out Di Maria's spot-kick after Lukasz Piszczek was adjudged to have handled Fabio Coentrao's cross.
Reus rendered that miss irrelevant in the 24th minute, taking advantage of Pepe's weak back-header to round Iker Casillas and slot the ball into the net.
Dortmund were rampant, Casillas tipping Mats Hummels' header over the bar from Reus' curling free-kick, before the Germany playmaker doubled his tally in the 37th minute as once again Madrid contributed to their own downfall.
Real made a bright start to the second half, Gareth Bale testing Weidenfeller with a stinging effort in the 49th minute, before a last-ditch challenge from Hummels denied Karim Benzema as the Frenchman prepared to shoot into an empty net on the hour mark.
