The hosts took a deserved lead before the break as Oscar de Marcos converted Markel Susaeta's cross in the 24th minute, but Abel Aguilar headed Depor level from Juan Carlos Valeron's driven ball just after half-time.
Athletic pressed for the winner but Fernando Llorente failed to find the target with three headers, with Andoni Iraola and Iker Muniain also coming close, and there was to be no way through for the Basque outfit.
With Aritz Aduriz suspended, Llorente was starting his first league game of the season and he came agonisingly close to opening the scoring in the 18th minute by bulleting Susaeta's free-kick off the bar and seeing his follow-up effort smothered on the line by German Lux.
Depor offered precious little as an attacking force in the first half, but they were nearly gifted a bizarre goal when Borja Ekiza badly misjudged a backpass to Gorka Iraizoz, with the defender lucky to see the ball dribble just past a post.
Athletic were dominant and deservedly took the lead midway through the half as Ander Iturraspe released Susaeta down the right, his cut-back met brilliantly by De Marcos who tucked home his fourth goal of the season.
All of Athletic's good work in the first half was wiped out within seven minutes of the restart, though, as a corner broke for Valeron on the by-line, with his cross being squeezed home at the near post by Aguilar's smart header.
The hosts then laid siege on the Depor goal and were unlucky not to retake the lead on a number of occasions.
Llorente headed over the bar twice when well-placed to score, whilst a driven cross from Iker Muniain somehow avoided Llorente and a number of Depor defenders facing their own goal when any touch would have taken the ball in.
A wayward cross from Jon Aurtenetxe then came within inches of dropping in with Lux stranded, but the Argentinian keeper made a good stop before parrying a Muniain drive moments later.
Llorente then saw another headed effort flash wide of the upright but, in a rare counter-attack, Depor almost stole all three points as Rodolfo Bodipo volleyed Javier Camunas' cross over the bar from just six yards out.