It was the home side who started the better, and they took the lead after 18 minutes when Javi Garcia found the bottom corner from inside the box.
It might have been 1-1 before the 30-minute mark, but Aduriz headed wide after good work from Inigo Perez.
The home side secured a two-goal lead, however, moments before the break when Joan Verdy finished off a sweeping move with a tidy finish.
Bilbao responded early in the second period as Aduriz headed home after he was found unmarked in the penalty area, while they were level with 20 minutes to play when Fernando Llorente found the bottom corner after smart work from Iker Muniain.
Espanyol moved back into the lead after 80 minutes, however, when Samuele Longo pounced on a loose ball to fire his side into the lead.
Longo saw red for a second yellow moments later, while Aduriz grabbed his second of the match to secure a point for Bilbao late on.