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.
Barcelona continued their unbeaten start to La Liga as the leaders cantered to a 4-0 victory at Levante.
Barcelona's Andres Iniesta was the star of the show, providing assists for Lionel Messi's quickfire double - his 81st and 82nd goals of 2012 - early in the second half.
Iniesta lashed home a goal of his own in the 57th minute before playing in Cesc Fabregas for a fourth six minutes later as Barcelona maintained their three-point lead at the top of La Liga with their 11th win in 12 matches.
Levante's Jose Barkero had the chance to pull a goal back for the hosts late on but saw his tame penalty saved by Victor Valdes.
Atletico Madrid stayed on Barca's coat-tails and stretched their 100% record at the Vicente Calderon this season to 10 games with a comprehensive 4-0 victory over nine-man Sevilla.
The game was almost over as a contest after just 20 minutes when Koke was brought down by Federico Fazio inside the area and - after the Italian defender had seen a red card for a professional foul - Radamel Falcao hammered home his 14th goal of the season from the penalty spot.
Two goals in four minutes just before half-time sealed the points for Atleti as Emir Spahic turned Arda Turan's cross into his own net before Koke capped a fine individual display with a stunning volleyed finish from Diego Costa's lobbed pass.
Sevilla's miserable night was rounded off when Ivan Rakitic received a second yellow card for hacking down Tiago, leaving Miranda to tap home a fourth goal in stoppage time.
Getafe heaped more misery on Espanyol by inflicting a 2-0 defeat on the league's bottom team at the Estadi Cornella-El Prat.
Pedro Leon's early strike set the visitors on their way and Mane put the result beyond doubt with a second goal in stoppage time.
Deportivo La Coruna moved out of the relegation zone thanks to a vital point earned from a 1-1 draw with Athletic Bilbao at San Mames.
The hosts took a deserved lead before the break as Oscar de Marcos converted Markel Susaeta's cross in the 24th minute, but Abel Aguilar headed Depor level from Juan Carlos Valeron's driven ball just after half-time.
On Saturday, Real Madrid's hopes of defending La Liga suffered a further dent on Saturday evening as they went down 1-0 at Real Betis.
Benat Etxebarria netted the game's only goal after 17 minutes as Pepe Mel's side went some way to redeeming themselves after last weekend's 5-1 derby mauling at the hands of Sevilla.
Malaga earlier registered their first league win in five games as they hammered Valencia 4-0 at La Rosaleda.
The margin of victory could have been even greater for the hosts, who controlled the game from the off and went ahead just eight minutes in when Francisco Portillo curled home from Eliseu's cut-back.
Javier Saviola made it 2-0 15 minutes from time - a goal that was quickly followed by a third from Roque Santa Cruz.
Valencia were then reduced to 10 men as Adil Rami saw a second yellow card and Isco rounded off a fantastic evening for Malaga with a fourth in stoppage time.
Elsewhere, Real Valladolid stretched their unbeaten home run to six games as Manucho's second-half strike secured a 1-0 victory over Granada.
The Angolan had come close to breaking the deadlock in the first half with an overhead kick but made no mistake just after the hour when he was played in by Oscar Gonzalez and sent a fierce shot beyond Tono into the top corner.
Finally, Rayo Vallecano left it late to see off out-of-form Real Mallorca 2-0 in an awful game on Saturday afternoon.
A match utterly devoid of excitement looked headed for a goalless draw until Leo Baptistao broke the deadlock with a goal totally out of keeping with what had gone beforehand, firing home low from 30 yards.
He then turned provider as substitute Andrija Delibasic got a second in the 90th minute to keep the hosts well placed in the top half and condemn the visitors to a seventh defeat in eight league games.
In Friday's game, Real Sociedad's recent good form deserted them as they were held to a 0-0 draw at home by struggling Osasuna.
Friday's La Liga Result
Saturday's La Liga Results
Sunday's La Liga Results
Monday's La Liga Fixtures