Aritz Aduriz ensured victory for Los Leones with a goal in the 33rd minute, heading home after Borja Oubina's misplaced clearance.
Athletic Bilbao have now recorded a record of five wins, seven losses and three draws this season, which puts them in 13th place, only five points above the relegation zone.
Celta Vigo are in 15th place, just two points above the relegation zone with a record of four wins, eight losses and three draws after 15 games.
En route to the win, Los Rojiblancos had the ball for 53.4% of the match.
Athletic Bilbao had 10 shots, including two on target.
Ibai Gomez had two shots for Los Leones, more than any of his team-mates.
Jonas Ramalho's 13 passes to Borja Ekiza were the most of any Athletic Bilbao team-mates during the match.
Markel Susaeta went off due to injury in the 88th minute and was replaced by Gaizka Toquero.
Celta de Vigo enjoyed possession for 46.6% of the loss.
Celta de Vigo had 11 shots, including one on target.
Enrique De Lucas made his mark on the match for Celta de Vigo after coming on for Natxo Insa in the 61st minute, as he took two shots.
- Sports & Recreation
- Athletic Bilbao