Atletico had the better of the play for most of the 90 minutes but paid for the profligacy in front of goal and their inability to find the final ball.
Martin Demichelis in the Malaga defence was marshalling Costa and Falcao well, although it was the final pass that was letting the away side down.
The match continued in the same vein after the half time, albeit with more of an effort to attack from the home side.
A couple of counter-attacks aside, though, it was still all Atletico, and they could have been ahead when Koke broke from midfield into the box, only to fire his shot wide.
Ten minutes later it was Cristian Rodriguez who was the culprit for the away side, miscuing from 20 yards when unchallenged, before Costa was booked for diving in the box. Atletico boss Diego Simeone responded by bringing on Adrian in his place.
With 15 minutes to go the home side finally created a goalscoring chance. It came from a good set piece, but Malaga captain Weligton headed wide from 10 yards out.
That chance sparked Atletico back into life as they upped the tempo and created two good chances, Falcao and Adrian both missed the target from good positions though.
Malaga survived a last-minute goalmouth scramble and held on to take a point from Atletico, who will be disappointed not to have won.