Sevilla were seeking revenge for their defeat by Espanyol in the King's Cup last week but they went behind to a Christian Stuani penalty after Salva Sevilla was felled by keeper Beto.
Sevilla, who are yet to lose at home this season, pushed forward and equalised in sensational style through a 35-metre drive from Diogo Figueiras after 26 minutes.
Vitolo slotted home from close range for Sevilla 10 minutes before the break and then Espanyol keeper Kiko Casilla was sent off as he handled a lob from Aspas outside the area.
Sevilla almost threw away the victory with Victor Sanchez firing home 15 minutes from the end but there was still time for Aspas, on loan from Liverpool, to hit the winner a minute from time.
Later second-placed Barcelona host an in-form Villarreal where victory would put them a point off leaders Real Madrid, who beat Real Sociedad 4-1 at home on Saturday.