Griezmann's shot from outside the box was then saved by Iraizoz.
Both sides continued to exchange chances, with Aritz Aduriz heading high from close range in the 33rd minute, and then Griezmann tried his luck again with a shot but in vain.
The Frenchman eventually made the breakthrough in the 62nd minute when he was sent through by Imanol Agirretxe and slotted home from close range.
Bilbao almost restored parity just three minutes later, as Aduriz forced Zubikarai into making a good save with a header from an Ibai Gomez cross.
However, the Basque giants' chances of a comeback disappeared in the 71st minute when they were reduced to 10 men.
Amorebieta handled the ball inside his own 18-yard box and was shown his yellow card, with Carlos Vela firing in the subsequent penalty.