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The hosts created the first decent opening of the match in the 20th minute when Bojan Jokic lofted a ball into a dangerous area, but none of his teammates was able to connect with it as the chance went begging.
They came closer still with eight minutes to go until the break as Giovani dos Santos went for goal with a free kick, sending his effort narrowly past the post.
The best chance of the half fell to the visitors, however, as Gabriel Paulista was penalised for a handball inside the area with just six minutes to go until the interval.
The hosts took full advantage of that let-off just two minutes after the restart when Oliver Torres skipped away from his man inside the box before sending a low pass across to Tomas Pina, who had the simple task of rolling it in at the back post.
Bilbao's hopes of getting back into the match were handed a boost in the 66th minute when Paulista was shown a second yellow card, reducing Villarreal to 10 men.
The visitors eventually made the most of their man advantage with seven minutes remaining when Ander Iturraspe sent a cross into the box that Aduriz met with a solid header into the bottom corner.
Bilbao had a good chance to claim all three points right at the end of the match as Aymeric Laporte found himself unmarked in a promising position, but his subsequent shot flashed just wide as the spoils were shared.