After a turbulent summer in which the club has sold several key players, Valencia were beaten by a side that only just survived relegation last season.
Maxi Gomez pulled Valencia level early in the second half only for Aspas to curl a superb free-kick into the top corner as Celta made it four points from their opening two matches.
"We had to hold on a bit at the end but we could have scored a couple on the break as well that would have made it more comfortable for us," said Aspas.
Valencia have not won away from home in the league since December.
"Away from home we have to be stronger," said Jose Luis Gaya. "They got through us too easily. We have to put it right because those errors have cost us."
Two of Valencia's rivals for the top four had both won earlier on Saturday as Getafe edged to a 1-0 win at home to Osasuna while Villarreal came from behind to beat Eibar 2-1.
Villarreal's Gerard Moreno cancelled out Kike's opener for Eibar before Paco Alcacer sealed victory for Unai Emery's team eight minutes later.