Two first half goals by Marco Reus, both coming from Real mistakes, gave Dortmund a deserved lead as the Spaniards, who had a penalty saved in the 17th minute, were left mesmerised by the Germans’ frantic pace.
Real, with top scorer Cristiano Ronaldo left on the bench due a minor knee injury, saw their 3-0 advantage from the first leg in Madrid almost evaporate in a disastrous first half for them.
Real, who twice lost to Dortmund in Germany in last season’s competition including a 4-1 thrashing in the first leg of their semi-final, briefly recovered in the second half before the Germans, last season’s runners-up, again surged forward, missing half a dozen clear scoring chances and also hitting the post.
The Germans got off to a fiery start, looking for an early goal that could set them on their way.
Real, however, squandered a golden opportunity to take the lead and put the tie all but beyond the hosts when they were awarded a penalty for a hand ball by defender Lukasz Piszczek and Angel Di Maria had it saved by Roman Weidenfeller.
A minute later it was Dortmund’s turn to miss with Henrikh Mkhitaryan firing wide from close range but Reus made amends in the 24th, pouncing on a defensive mistake by Pepe to slot in for the lead.
Mats Hummels almost added another as the hosts piled on the pressure but his well-timed header was palmed over the bar by Real keeper Iker Casillas.
Germany international Reus crashed in his second goal in the 37th when he charged forward after another Real mistake, this time from Asier Illarramendi, fed striker Robert Lewandowski who hit the post, and was left unmarked to drill in at the far post.
Real brought on Isco in the second half to control possession and they briefly succeeded as Dortmund lost some of their passing sharpness and pace.
Dortmund were lurking on the break and almost levelled the tie when Armenian Mkhitaryan rounded Casillas but only managed to hit the post in the 65th minute.
The Germans then poured forward, determined to score a third that would force extra time but could not find the back of the net, leaving the competition with their heads held high.
(Reporting by Karolos Grohmann; Editing by Rex Gowar)