Mario Gotze was the guiltiest party for Dortmund as he missed three golden opportunities and Robert Lewandowski fluffed the best opening as Malaga coach Manuel Pellegrini saw his team achieve his pre-match objective of not conceding an away goal.
Dortmund were the better team in the first half and should have taken the lead on 14 minutes when Lewandowski's flick-on sent Gotze clear, but he failed to connect cleanly with his shot, which was pushed away by Willy Caballero.
Malaga keeper Caballero denied Gotze again five minutes later, using his legs to stop the Germany international's shot after a clever pass from Ilkay Gundogan.
Kevin Grosskreutz rolled the ball into the net, but Lewandowski was flagged offside on 24 minutes before Marco Reus was the next to bring a save out of Caballero.
The home side finally came into the game as an attacking force in a five-minute spell before half-time, with Dortmund keeper Roman Weidenfeller fortunate not to be punished after mishandling Joaquin's cross.
Joaquin's corner on 43 minutes saw Weidenfeller scoop away Vitorino Antunes' header and Jeremy Toulalan's attempt from the rebound was cleared away from the post by Gotze.
But Dortmund tested Caballero again before the half-time whistle sounded as the Argentinian thrust out a hand to turn away Sebastian Kehl's swerving 25-yard effort.
Within two minutes of the restart, Dortmund had another clear-cut chance to score, but Lewandowski uncharacteristically mis-kicked 12 yards out after Gotze's pull-back had deflected into the reported Manchester United target's path.
Dortmund continued to create openings and Gotze was played in again by another superb pass on 65 minutes only to roll his shot across the face of goal and beyond the far post.
Malaga responded as the highly-rated Isco's fierce shot from the edge of the area was beaten away by Weidenfeller, who reacted smartly to keep out the follow-up from Toulalan.
Substitute Julian Schieber was afflicted by the same lack of composure as his team-mates when he shot wildly off target after a sharp turn inside the box and Dortmund will hope they do not rue their first-leg profligacy in next week's return meeting in Germany.