Bilbao looked lively when they had the ball, with Artiz Arduriz looking to put the Barcelona back four under pressure while running at their goal, but it was cleared before he could get his shot away.
Munir thought that he had broken the deadlock at the start of the second half after turning a low cross from Messi into the back of the net, but the assistant referee ruled him offside.
Jordi Alba fired a powerful effort towards the bottom corner as the hosts continued to press for a goal, but Iraizoz got down to make the stop.
The defender was involved again later in the half as he tried to pick out Messi with a cross from the left, but Aymeric Laporte beat the Argentine to the ball to clear the danger for Bilbao.
Neymar came off the bench inside the final half hour, crossing the ball towards Messi, but the striker sliced his finish high from close range.
The Brazilian managed to open the scoring himself with just over 10 minutes remaining, collecting a through ball from Messi before picking out the bottom corner.
Just five minutes later, he added his second following another assist from Messi, putting the win beyond doubt.