Five minutes into the second half, the hosts were awarded a penalty when Aduriz latched on to a loose ball inside the box and was brought down by Mikel Gonzalez, who was subsequently shown a red card for his challenge.
Aduriz himself stepped up to take the spot kick and he placed a composed finish into the bottom corner to give his side the lead.
Within minutes the home side almost grabbed a second goal when Aduriz was picked out in the middle by Andoni Iraola, before volleying an acrobatic effort over the bar.
However, on the hour mark Sociedad were back on level terms, as Ruben Pardo slid a ball through to Alberto de la Bella, who took a touch before curling a strike into the far corner and beyond the reach of Gorka Iraizoz.
Iraola then came close to restoring Bilbao's lead with a powerful strike from the edge of the box that went inches wide of the post.
Despite their numerical disadvantage, Sociedad continued to look a danger and Carlos Vela showed his threat with a direct run from inside his own half that ended with a wayward shot from the edge of the box.
Vela was involved again minutes later, as his fierce volley on the turn was tipped over by Iraizoz.
At the other end of the pitch, Inaki Williams was inches away from scoring for the hosts, when his curling strike from the edge of the box went just wide of the target.
Bilbao continued to press for a winner and they put their opponents under intense pressure, but they were unable to find a way past Rulli for a second time as the match finished on level terms.