The visitors almost made a perfect start to the match when Aritz Aduriz found space inside the box in the very first minute, but his strike flashed inches wide of the left post.
Ander Herrera came even closer to breaking the deadlock in the 18th minute when he found space to shoot from the edge of the box, only to see his strike crash against the upright.
Bilbao were handed a good chance to equalise just six minutes after the restart when Feghouli gave away a penalty. Aduriz stepped up and made no mistake with his spot kick to level the score.
The hosts could have regained the lead in the 64th minute when Seydou Keita latched onto a cross inside the six-yard box, only to put his effort wide to the left.
Neither side could find a winner in the closing stages, however, as the spoils were shared at Estadio de Mestalla.