Bilbao took the lead with just three minutes played when Markel Susaeta received the ball on the edge of the box and hammered a shot into the top corner.
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However, Osasuna came back quickly and were level through Emiliano Armenteros's free kick.
Iker Muniain took too long to shoot and was closed down as the away side searched for a third goal before the break.
Moreno Mikel Rico then curled wide early in the second half and Susaeta fired a free kick just wide of the post. The home side had two decent chances to equalise, but Damia and Marc Bertran missed the target when well placed.
Aduriz then got his second of the game with just over an hour played as he converted a deep cross at the far post.