Aspas got the only goal in the 83rd minute as Celta picked up not only their first victory but their first points on the road this season.
The win moves them three points clear of the bottom three and to within a point of their opponents.
Helder Postiga, mysteriously dropped for the weekend defeat against Barcelona after scoring four goals in his previous four games, did his best to return with a bang for Zaragoza.
He tested Javi Varas from range with just 34 seconds on the clock, but that was hardly a sign of things to come from the hosts.
Celta Vigo did most of the early probing, but decent openings were in short supply.
Gustavo Cabral's shot was charged down by Glenn Loovens while Aspas sent an effort wide and Borja Oubina had a chance go begging when he failed to make proper contact on a shot.
That about summed up a poor first-half all-round, but the entertainment improved after the break.
Again Zaragoza had early chances, with Victor Rodriguez's long-range effort parried by Varas and Postiga shooting straight at the goalkeeper seconds later.
Cabral fired wide at the other end while Mario Bermejo brought a better save out of Roberto Jimenez with a close-range shot.
Jimenez was busy again in the 70th minute, pushing clear a shot from Aspas as Celta got closer to a goal.
It eventually arrived in the 83rd minute, with Roberto Lago playing the ball in from the left for Aspas to slot home.
Carlos Aranda sought an immediate response for the hosts, but Varas tipped his shot over the crossbar.
There was an urgency to Zaragoza now, but it had come too late and when Loovens headed wide from a corner their final chance had gone.