The hosts had more chances, but were unable to breach the visitors' defence.
Alaves were bidding to respond from a 3-0 loss at Athletico Madrid, while Athletic were seeking to build on the win over Girona last time out.
Jonathan Calleri failed to convert Ibai Gomez's cross early on before Raul Garcia had an effort saved for the visitors.
Calleri headed Ximo Navarro's cross wide before Gomez was denied by Iago Herrerin.
Fernando Pacheco denied Garcia and Aritz Aduriz fired wide for Athletic.
Inigo Cordoba shot off target for the visitors in the second half before having another effort saved.
Jony had a shot saved by Herrerin, Ruben Duarte shot wide and Guillermo Maripan headed wide as Alaves spurned three chances.
Ruben Sobrino and Calleri also shot off target as the hosts failed to break the stalemate.