The German outfit trailed 3-0 from the first leg last week, but had real hope of turning the tie around when Marco Reus scored twice in the first 45 minutes.
However, despite a number of second-half chances for the home side, Madrid held on to book their spot in the final four of the tournament.
Here, Sports Mole dissects the 90 minutes of action between the two European rivals.
On target: 6
On target: 5
Was the result fair?
There is no question that Dortmund deserved to win on the night, but ultimately, they have exited the competition. The home side were quicker, stronger and more inventive than Madrid, but could not find that elusive third goal, which they certainly deserved. A number of chances also arrived in the second 45 minutes, but an inspired performance from Madrid skipper Iker Casillas ensured that Los Blancos made it into the semi-finals for the fourth successive season. Madrid had their chances in the latter stages and indeed passed up a first-half penalty, but Dortmund were well worth their clean sheet.
In his pre-match press conference, head coach Jurgen Klopp insisted that if his side were to exit to Madrid this evening, then they would go down fighting. Well, that was a promise that the German certainly kept. An uneventful opening 15 minutes produced very few chances for either side, but Madrid had a penalty after 16 minutes when Lukasz Piszczek was penalised for handball inside his own box. Angel di Maria missed from the spot, however, and from that moment, Dortmund dominated. Reus's first-half double left the German side 2-0 ahead at the interval and they had a number of terrific chances in the second 45 minutes. Ultimately, however, the opportunities fell to the wrong player, with midfielder Henrikh Mkhitaryan guilty of missing a trio of golden chances. It was an inspired performance from the German outfit, who were also defensively brilliant, but they just could not find that third strike in front of a capacity crowd.
Madrid might have advanced into the semi-finals of the Champions League this evening, but head coach Carlo Ancelotti will take few positives from this particular match. His side were out-fought and out-played in Germany and but for some brilliant goalkeeping from Casillas and poor finishing from Dortmund, could and probably would have exited the competition. Without a certain Cristiano Ronaldo, who was only fit enough for the bench, Los Blancos lacked a real focus point in attack and too often lost possession in the middle of the park. Xabi Alonso and Luka Modric have both enjoyed brilliant seasons, but they also struggled this evening under constant pressure from the home side. Centre-backs Pepe and Sergio Ramos were also both caught out on a number of occasions as Dortmund squeezed the play in forward areas. It might have been a different night entirely had Di Maria converted from the penalty spot in the early stages, but the Argentine's effort was saved by Roman Weidenfeller. Their hopes of a 10th European Cup remain alive, but their performance this evening would not have worried their rivals for the crown.
Sports Mole's man of the match
Marco Reus: There were a number of contenders for this award this evening, but Reus simply must be handed the prize. The German international had a stunning first period - with his two strikes moving Dortmund to within one goal of Madrid in the tie. Los Blancos just could not deal with Reus's pace and movement as the attacker also twice set up Mkhitaryan in the second period, but the Armenian could not convert. It was an inspired performance from the 24-year-old, who has scored 20 times in 37 games for Dortmund this season.
There were a number of incidents that could have been recognised for the biggest gaffe of the evening - including a poor backwards header from Madrid defender Pepe in the lead-up to Dortmund's first - but Mkhitaryan's trio of misses cost his side. The attacking midfielder had a terrific chance in the second period after collecting a through pass from Reus before rounding Casillas, but somehow he struck the post from close range. That incident occured after he had fired wide from inside the Madrid box and before he struck straight at Casillas from close range. Mkhitaryan scored 29 times for Shakhtar Donetsk last season, but has managed just nine for Dortmund this season.
It was certainly an eventful evening for referee Damir Skomina. The Slovenian was extremely lenient in the first period, with Madrid midfielder Xabi Alonso just one player lucky to escape a card on a number of occasions. Dortmund will also feel that Alonso should have seen red in the second half for a late challenge on the heavily-involved Mkhitaryan, but Skomina waved play on. The penalty award was perhaps harsh, but replays did show that Dortmund right-back Piszczek handled Fabio Coentrao's cross inside the box. That said, overall, Skomina did allow the play to flow in the correct areas and contributed to what was a free-flowing evening of football.
Dortmund: Klopp's side return to Bundesliga action away to champions Bayern Munich on Saturday.
Madrid: Los Blancos also return to their league duties on Saturday with a home clash against Almeria.