Athletic strikers Iñaki Williams and Oihan Sancet struck either side of half-time.
"We're happy. We've come here, where there's a tough atmosphere, played well and got a result," Valverde said post-match. "But now we have to keep this up. Thirteen points out of 18? That tells me we've dropped five on the way."
"It was a derby, we played well. Both teams were strong and little by little we took control of the match."
Valverde added: "We conceded very little in the first 85 minutes. We always want to keep a clean sheet and give away as few chances as possible. If we maintain our intensity from start to finish we're a tough to beat. At the moment we're resolute, which is what we're looking for."