Midfielder Benat earned Athletic all three points in fine style when he exquisitely curled home from the edge of the area early in the second half.
The victory moves Ernesto Valverde’s side level on points with seventh-placed Malaga, who occupy the final Europa League play-off place.
Cordoba have failed to win any of their last 15 league games and are eight points adrift of safety with five matches remaining.
Both sides made tentative starts and Athletic defender Mikel San Jose and Cordoba midfielder Fede Cartabia failed to test the respective goalkeepers with long-range strikes inside the opening 15 minutes.
Cordoba’s on-loan forward Bebe very nearly put his side into the lead shortly after the half-hour mark but his shot from a difficult angle could only crash into the crossbar and rebound to safety.
Athletic striker Aritz Aduriz did well to rise between two Cordoba defenders in first-half stoppage time but his header was well saved by Cordoba goalkeeper Juan Carlos, who needed to be at his athletic best to keep his goal intact.
The second half was just four minutes old when Fidel wasted a glorious chance to put Cordoba into the lead when he blazed over the crossbar from close range and the hosts were left to rue that missed opportunity when Benat struck a few minutes later.
Cordoba needlessly gave away possession in their own half and San Jose’s subsequent one-touch pass found Benat, who cut inside and unleashed a ferocious shot that nestled into the net before Juan Carlos could ready himself.
The hosts stepped up the pressure in the latter stages but on-loan duo Rene Krhin and Aleksandar Pantic could not find a way past Gorka Iraizoz and Athletic held on for their first win in four matches.