Celta had the first chance of the game as Theo Bongonda cut inside from the right-hand side, but the strike from outside the box was put wide.
It would not be long before Aspas got another chance as he ran at the Espanyol defence, took on one player and then chipped the ball over a helpless Pau Sabata for the opener just before half time.
In the second half, Pablo Hernandez was sent off just two minutes after the restart for a second bookable offence, leaving Celta with 10 men for the remaining 43 minutes.
Aspas had another chance after being set up by Orellano just outside the box but the strike was palmed away by Sabata onto his own bar and away.
The former Liverpool man had yet another opportunity to put Celta further ahead after he rounded Sabata one on one but he could not place his finish into the open goal.
The home side held on to move into fourth in the La Liga table, while Espanyol remain 12th.