Aspas got the winner in the 62nd minute with a clever redirection of a free kick from Lucas Olaza and found the far corner of the goal.
The hosts won despite playing most of the match with 10 men because of a straight red card shown to Croatian midfielder Filip Bradaric for a hard foul in the 21st.
The victorylifted Celta out of the relegation spots. The Vigo team is now two points clear of 18th-place Mallorca, which drew 3-3 at Real Betis on Friday.
Leganés stayed second-to-last with the loss, tied on points with last-place Espanyol, which visits Valladolid on Sunday.
Leganés midfielder Óscar Rodríguez was shown a red card just before Aspas’ goal, but it was waived off after a video review.
The home victory for Celta followed a 2-2 draw against leader Real Madrid at the Santiago Bernabéu Stadium last weekend. It had beaten fifth-place Sevilla at home before that.
Leganés has won only one of its last seven league games. The modest club from southern Madrid lost its main forward this week, Martin Braithwaite, in a transfer to Barcelona. Leganés was not given permission to replace him outside the transfer window. Barcelona was able to make the signing because of an injury to Ousmane Dembele.
Leganés, scoreless in three league matches,next hosts Alavés, while Celta visits Granada.