The Germans scored twice in stoppage time, including a controversial winning goal from Felipe Santana, that saw them complete a sensational comeback and advance to the last four.
Dortmund players in front of the Malaga goalkeeper Willy Caballero looked offside when the ball was crossed for the winning goal and Santana also looked offside when he scored.
The Germans had looked on their way out in the 82nd minute when Eliseu scored to put Malaga 2-1 up.
Dortmund, needing two goals to advance after their goalless draw in Spain last week, pulled one back through Marco Reus a minute into stoppage time to rekindle their hopes.
Santana then scrambled the ball over the line from what looked an offside position to complete a dramatic night.
"We are still trying to take in what has happened. The dressing room is very depressed," Malaga striker Roque Santa Cruz told Spanish television.
"It was a game that incredibly we let slip out of our hands and in the end they had a lot of luck. Not only for scoring goals but because of the situations as well. The third was offside. They had a lot of luck."
"We are very disappointed. We were four minutes from a semi-final and everything slipped out of our hands in those last few minutes," he told Spanish television.