It was the hosts who created the first opening of the encounter at San Mames as Mikel Balenziaga drilled in a teasing low delivery that no teammates could get on the end of to tap home.
Inaki Williams began to threaten for the hosts in the closing stages of the first half, putting one effort into the side-netting moments after he had almost turned home an enticing cross from Ibai Gomez.
Bilbao continued where they left off at the start of the second half and should have broken the deadlock just minutes after the restart, only for Diego Alves to deny a header from an unmarked San Jose.
An end-to-end period of the match then saw Nicolas Otamendi almost break the deadlock in bizarre fashion as his long through-ball was misjudged by Iraizoz and bounced narrowly wide of the post.
The Argentine defender was in action again seconds later as he cleared Aduriz's header off the line with Alves beaten.
The visitors made the most of that let-off by taking the lead shortly after the hour mark when substitute Rodrigo De Paul finished with aplomb following slick build-up play from Rodrigo Moreno and Alcacer.
Rodrigo then had three chances to give his side a cushion, but he missed the target on the first occasion before being thwarted twice by Iraizoz in quick succession.
Valencia's hopes of holding on took a blow with eight minutes remaining when Otamendi was shown a red card for a high challenge on San Jose, prompting a scuffle between the two sets of players.
Bilbao applied the pressure late on and were rewarded with a scrappy equaliser as Aduriz applied the finishing touch after Borja Viguera had seen his shot come off the upright.
The result sees Valencia move one point clear of Sevilla in the race for a Champions League place, while Bilbao remain eighth. body check tags ::