Bilbao midfielder Oscar de Marcos fired the Basques ahead in the 43rd minute at Celta's Balaidos stadium when he crowned a sweeping move with a handsome finish past goalkeeper Javi Varas.
The visitors squandered several chances to kill off the game and striker Aritz Aduriz somehow skewed a close-range shot wide when he had the empty goal at his mercy with under half an hour left.
Aspas conjured a goal from nothing six minutes from time when he lashed a low angled shot between two defenders that flew just inside the post.
After reaching the final of the Europa League and the King's Cup last term, Bilbao have had a woeful season and Friday's draw leaves them down in 14th place on 38 points with four games left.
Financially troubled Celta, promoted at the end of last season after five years in the second division, are 18th on 31 points having taken seven from their last three outings.
The two teams below them in the relegation places, Real Zaragoza and Real Mallorca, are only one and three points behind respectively and play at home to Rayo Vallecano and Levante on Sunday.
Galicia-based Celta have fallen a long way since they finished fourth in the 2002-03 season and qualified for the Champions League.
They lost 5-2 on aggregate to Arsenal in the last 16 the following term but despite outperforming in Europe's elite club competition, they finished 19th in La Liga and were relegated.
Leaders Barcelona have an 11-point lead over last season's champions Real Madrid in second and need five points from their remaining five games to secure a fourth title in five years.
They host Real Betis on Sunday and if Real Madrid fail to win at home to Real Valladolid on Saturday, a Barca victory would be enough as they would be at least 13 points ahead with only 12 left to play for.