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Aduriz should have doubled his side's lead and his own personal tally midway through the half as he latched on to a cross inside the box, but this time he put his effort wide of the target.
There was no way he could miss at the start of the second half, however, as he tapped home from close range to make it 2-0 just 33 seconds after the restart.
Malaga's first real opening came 10 minutes later when Sergi Darder nodded a cross towards goal, only to see his header go just wide.
Aduriz almost had a hat-trick just before the hour mark when his first-time effort drew a fine stop from Wilfredo Caballero in the Malaga goal.
The hosts wrapped up the points just four minutes later, however, getting their third of the night as Caballero allowed Ander Herrera's mis-hit volley from the edge of the box to squirm past him and over the line.
Malaga had a good chance to get a consolation goal 15 minutes from time when Roque Santa Cruz was found with a cross into the box, but he planted his header well wide.
Bilbao could have added insult to injury late on as Iker Muniain was played through on goal, but some fine defending denied the hosts a fourth.