Iraizoz could do nothing to stop Adriano's effort towards the bottom corner midway through the first half, but Mikel Balenziaga was in the right place at the right time to clear the ball off the line.
Barca came close again in the 26th minute as they threatened from a corner, but this time Marc Bartra couldn't get the finishing touch.
Bilbao's first sight of goal came shortly after the half-hour mark as they came within inches of breaking the deadlock in spectacular fashion through an Aritz Aduriz bicycle kick that struck the inside of the post.
The hosts responded by hitting the woodwork themselves moments later as Pedro fed the ball to Alexis Sanchez, who hit the crossbar when he should have scored.
Sanchez and Pedro both threatened again five minutes before the interval, but Iraizoz got down well on both occasions to keep the scores level going into the break.
The visitors took the lead five minutes after the restart when Bartra's mistake allowed Aduriz to drive into the box and fire an effort past Pinto and into the bottom corner for his eighth goal in the last 10 games.
Barcelona went looking for an equaliser but once again found Iraizoz in fine form as he denied Pedro from inside the box after Xavi had created the chance for his teammate.
The hosts spurned another glorious opportunity moments later as Messi raced through on goal following a pass from Sanchez, but the Argentine slotted his effort past the post.
Bilbao almost doubled their advantage just a minute later when Ander Herrera let fly from range, only to see his effort clip the top of the crossbar on its way behind.
Barca finally got the equaliser in the 72nd minute as Pedro reacted quickly to turn a Sanchez effort past Iraizoz from close range.
Messi came close again before the quickfire turnaround was completed two minutes after the equalising goal, with the Argentine firing a low free kick into the bottom corner.
Bilbao almost hit back with just 10 minutes to go, however, as Markel Susaeta drove into the box, only to fire his shot straight into the arms of the keeper.
Cristian Tello squandered a fine chance to seal the victory for the hosts in stoppage time before Javier Mascherano made brilliant late block to deny Bilbao an equaliser as Barca moved back up to second in the table.