Bilbao were the side making most of the running in the early stages and they came close to breaking the deadlock when Mikel San Jose thumped a header against the crossbar from Oscar de Marcos's corner.
The hosts tested soon after the half-hour mark when Michael Krohn-Delhi whipped a free kick over the wall, forcing Gorka Iraizoz into a diving save.
Minutes later they came even closer when Alex Lopez flicked a looping header towards the far post and it bounced on top of the crossbar.
Just before the interval it was Bilbao who took the lead through De Marcos, who ran onto Susaeta's through-ball and clipped it over Javi Varas.
Midway through the half the woodwork was rattled for a third time by Athletic's Ibai Gomez, who let fly from distance only to see his effort cannon off the crossbar.
Two minutes later it really should have been game, set and match. Iker Muniain squared for Aduriz to tap into an empty net, but the forward contrived to miss the target from a few yards.
The visitors were made to pay for their profligacy with six minutes on the clock when Celta's top scorer Aspas fired hard and low into the bottom corner.
A point moves Celta out of the bottom three on goal difference while Athletic are seven clear of the drop zone in 14th.