By FOX Sports staff
Joaquin opened the scoring for the visitors at the Signal Iduna Park in the 25th-minute with a well-taken strike, but Robert Lewandowski equalised four minutes before the break when he applied the finishing touches to a brilliantly-worked move.
Even if the score had remained at 1-1, Malaga would still have advanced to the semi-finals on the away goals rule, after last week's first leg in Spain had ended in a 0-0 draw, but the La Liga outfit looked to have made sure of their progress in the 82nd minute when Eliseu netted their second.
But in a dramatic end to the tie, Dortmund pulled level in the second minute of injury-time when Marco Reus lashed home a rebound, and a minute later, Felipe Santana bundled the ball over the line after a goal-mouth scramble to send his side through in dramatic fashion.
Here, FOX Sports breaks down where the game was won and lost on Tuesday evening.
For the second week running, Roman Weidenfeller played an important role for BVB as he made a number of excellent saves early on to keep his side in the game.
Marcel Schmelzer overcame a slow start to provide plenty of energy down the left, and his marauding runs were key to Dortmund's strong showing in the second half. However, the same cannot be said for Lukasz Piszczek, who struggled to get going.
Despite scoring what proved to be the winner, Santana had a below-average game as he caused his team-mates unnecessary duress with his hesitant defending, although Neven Subotic was regularly on hand to bail him out.
Weidenfeller - 7.5 | Piszczek - 6.5 | Subotic - 7 | Santana - 6.5 | Schmelzer - 7
Dortmund really missed the suspended Sebastian Kehl early on as Sven Bender struggled to fill his captain's shoes, which allowed Malaga plenty of space in the middle of the park.
Ilkay Gundogan, on other hand, got off to a promising start as he expertly linked defence with attack, but he gradually faded out of the game.
Mario Gotze was however the star of the show for Dortmund and proved again why he's regarded as one of the hottest talents in the world. The 20-year-old caused all kinds of problems for the opposition defence with his skill on the ball and uncanny knack of finding space in the attacking third, and was involved in most of his side's meaningful attacks.
Bender (off 72') - 5 | Gundogan (off 86') - 7 | Gotze - 8.5
Unlike last week when he struggled to have an impact, Lewandowski was back to his best on Tuesday and cancelled out the visitors' opener in the 41st minute when he applied the finishing touches to a fantastic team move.
Reus was also integral in that goal as he sent the Polish striker through with an audacious back-flick, before getting Dortmund's second equaliser when he reacted quickest to a loose ball and fired into the bottom corner.
Jakub Blaszczykowski was given the nod ahead of Kevin Grosskreutz but failed to really have an influence, despite providing a couple of dangerous runs in the opening 45.
Blaszczykowski (off 72') - 6 | Reus - 7.5 | Lewandowski - 8
Schieber offered the hosts plenty of drive down the right after coming on and nearly laid on a goal for Reus with a pinpoint low cross, only for Willy Caballero to come to Malaga's rescue.
Sahin's forgettable season continued as he failed to influence proceedings after coming on in the 72nd minute, while Mats Hummels had little time to make an impact after replacing Gundogan with four minutes to go.
Schieber (on 72') - 6.5 | Sahin (on 72') - 6 | Hummels (on 86') - N/A
Had it not been for Caballero, Malaga would probably have been out of the tie way before Santana's 93rd-minute winner. The veteran goalkeeper made a number of brilliant saves and was unlucky to be on the losing side.
Likewise, Martin Demichelis was an undeserving loser as he was a rock at the back, but Sergio Sanchez's inclusion, coming in for the suspended Weligton, proved costly as he was exposed time and time again by the pacy opposition forwards.
The ever-improving Antunes nearly singlehandedly denied Dortmund their winner after blocking shots from Santana and Schieber, only to see the former stab the ball home eventually.
Down the other flank, Jesus Gamez proved useful both in defence and attack, especially considering he spent most of the evening up against Reus.
Caballero - 8.5 | Gamez - 7 | Demichelis - 7.5 | Sanchez - 5 | Antunes - 7
Rolling back the years again, Joaquin handed Malaga the perfect start when he skipped past Schmelzer on the edge of the area, before drilling a shot between the legs of Subotic and into the bottom corner. Apart from the goal, the ex-Spain international always posed a threat with his pace and composure on the flanks.
Duda also tried his best to get the visitors across the line and carved out a plethora of chances with his set-piece deliveries, while Jeremy Toulalan, although not as impressive as he was in the first leg, provided plenty of drive and bite in the engine room.
Nonetheless, the selection of Ignacio Camacho backfired as the youngster, once regarded as the next big thing in Spanish football when he broke onto the scene at Atletico Madrid, was anonymous in central midfield, and gave away possession cheaply a number of times.
Joaquin (off 87') - 8 | Toulalan - 7 | Camacho - 5 | Duda (off 74') - 7
Like Gotze, Isco is currently regarded as one of the brightest prospects in world football but he once again failed to perform on the big stage. He did initiate the move that led to Duda's goal, but apart from a few moments of brilliance, the 20-year-old seemed to be avoiding the ball.
Julio Baptista did his best to make up for his strike partner and held up possession well in the attacking third, and also got an assist when his goal-bound shot was helped over the line by Duda.
Isco - 6 | Baptista (off 84') - 6.5
Eliseu looked bright after coming on and looked set to become Malaga's hero when he netted to make it 2-1, although it was not to be in the end.
Roque Santa Cruz and Francisco Portillo were both brought on late in the game and did not exactly have much time to influence proceedings.
Eliseu (on 74') - 7 | Santa Cruz (on 84') - N/A | Portillo (on 87') - N/A