Real Madrid survived a huge scare to progress through to the Champions League semi-finals as Borussia Dortmund fell just short of a stunning second-leg fightback.

Dortmund had it all to do after losing the first leg 3-0 in Spain but last season’s runners-up looked on course to stage a remarkable comeback when, after seeing Madrid winger Angel Di Maria miss an early penalty, they scored twice through Marco Reus during a first-half onslaught.

The German side continued to throw everything at Madrid in the second half, hitting the woodwork and going close on a number of other occasions, but they could not find the third goal they needed as Carlo Ancelotti’s men held on for a 3-2 aggregate win.

Any optimism in the home camp should have been extinguished in the 16th minute when Madrid won a penalty after Fabio Coentrao’s cross struck the arm of Lukasz Piszczek. Di Maria stepped up for the penalty but he slipped just as he shot and saw his effort well saved by Roman Weidenfeller diving to his left.

The miss galvanised the home side and they laid siege to the Madrid goal for the remainder of the half. The unmarked Henrikh Mkhitaryan should have scored in the 19th minute when he side-footed wide following good work by Robert Lewandowski, but within five minutes they were ahead thanks to a combination of Reus and a terrible error from Pepe.

The Portugal centre-back’s attempted back header was pounced on by Reus and he nipped it past Iker Casillas before sliding it into the net.

Casillas did well to tip over Mats Hummels’ powerful header at the far post in the 32nd minute but Dortmund would not be denied and they made it 2-0 in the 37th minute. Asier Illarramendi carelessly gave the ball away to Reus, who raced to the edge of the Madrid area before slipping in Lewandowski. Casillas tipped the Poland international’s shot onto the post but Reus followed up.