Ernesto Valverde's men could have gone six points clear of fifth-placed Villarreal following their 4-2 defeat to Real Madrid, but were left to settle for a share of the spoils at Estadio Balaidos.
The hosts dictated play for much of the first half, with winger Santi Mina and striker Charles both going close to opening the scoring
Indeed, Bilbao rode their luck at times in the opening period and would have fallen behind had Charles' rasping effort not hit the post just before half-time.
Bilbao came into the game more in the second half and could have snatched the points 12 minutes from time when Iker Munain curled an effort narrowly wide.
It was Celta who continued to enjoy the better of the opportunities and should have sealed a third straight league win in the final minute of time added on as substitute Mario Bermejo headed just over the crossbar from Fabian Orellana's cross.
The draw keeps Celta in 11th place on 26 points, while Bilbao sit four points ahead of Villarreal in the final UEFA Champions League spot.
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