Malaga appeared to be going through to the last four in their sparkling debut season in the Champions League but their dreams were dashed when last-gasp goals from Marco Reus and Felipe Santana sent the Germans through in a stunning encounter.
Santana's winner left the Malaga players fuming because when the cross came in four Dortmund players appeared to be offside and the Brazilian defender himself also looked offside when he poked the ball over the line in the dying seconds.
"Football is an act that involves many factors. The third goal was an offside goal and there were all kinds of elbows and punches," said Malaga coach Manuel Pellegrini.
For Dortmund, European champions in 1997 but close to financial ruin in 2005, it was another big step on a long road back to success after they lifted the Bundesliga title in the previous two seasons.
"I have never experienced anything like this. We were already dead," Dortmund CEO Hans-Joachim Watzke told reporters.
The club's sports director Michael Zorc added: "It is difficult to put this into words. We were lucky tonight. The team missed so many chances but kept on believing."
Malaga had a difficult run-up to the match, losing in La Liga on Saturday and travelling to Germany without Pellegrini, who arrived early on Tuesday after returning to his native Chile at the weekend following the death of his father.
LEWANDOWSKI SCORES
The Spaniards initially tried to stifle Dortmund's game and pressed high up the pitch to stop them launching quick attacks.
Dortmund gradually found their passing game and came close to scoring when Robert Lewandowski chipped the ball over goalkeeper Willy Caballero but it sailed just wide.
However, Malaga, who were held to a 0-0 draw in the first leg, stunned the 66,000 crowd when Joaquin shook off his markers and drilled in a low shot for the lead after 25 minutes.
Dortmund striker Robert Lewandowski drew the hosts level in the 40th and they squandered a string of chances in the second half through Reus and Mario Gotze, much like in Spain where Dortmund failed to score despite dominating.
Eliseu appeared to seal Dortmund's fate when he put the Spaniards ahead in the 82nd minute, although he looked offside when he received the ball inside the area.
The Germans refused to give in, though, and sparked wild celebrations among their fans by booking a semi-final place.
Malaga, who toppled former European champions Porto in the last round and AC Milan in the group stage, had looked set to become the first team to beat Dortmund in Europe this season but ended up feeling robbed of a spot in the last four.
"We are very disappointed. We were four minutes from a semi-final and everything slipped out of our hands in those last few minutes," said Malaga striker Roque Santa Cruz.
