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.
"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."
The Spaniards initially tried to stifle Dortmund's game and pressed high up the pitch to stop them launching quick attacks.
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.
"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.
