Celta Vigo looked threatening in the opening minutes but the home side soon got in to the game and dominated early possession.
The visitors could have had a penalty after Fernando Amorebieta appeared to handle the ball in the area, but the referee waved away Bilbao's appeals.
Bilbao had several chances for a second before half-time, but Varas prevented them extending their lead.
Both sides showed more intent after the break as Bilbao looked to secure the win, and Celta searched for an equaliser.
Despite the increase in tempo, there were few attempts on goal, and the game finished 1-0.
The win moves Bilbao up to 13th, while Celta Vigo are 15th, just three points above the relegation places.