Joseba Llorente headed onto the bar with the visitors' best effort early on, but thereafter the hosts dominated and found the breakthrough when Augusto Fernandez's cross was flicked home by Aspas 21 minutes from time.
Alejandro Arribas was then dismissed for a second bookable offence and Celta made their man advantage count in stoppage time when Bermejo rolled in the second.
Celta's transfer deadline day capture Park Chu-young was not included by Paco Herrera but Osasuna did hand a debut to Ghanian midfielder Anthony Annan in place of suspended captain Patxi Punal.
And the former Rosenborg man's combative presence was influential as Osasuna made a positive start.
Roland Lamah's swinging cross from the left was met perfectly by the diminutive Llorente on 10 minutes but the ball looped onto the top of the crossbar.
Celta had been outclassed in losing to Real Sociedad last weekend but their confidence began to return as the half wore on and Andres Fernandez had to make a fine one-handed save to turn away a powerful header from Aspas on 25 minutes.
Sisi had a decent effort well held by Javi Varas just after the break, but the hosts controlled the second period and went in front on 69 minutes when Fernandez twisted and turned on the right before delivering a perfect cross for Aspas to head in is first goal of the season.
Any hope of an Osasuna comeback was then extinguished seven minutes from time when Arribas was rather harshly shown a second yellow card for a push on Bermejo and Celta made sure of the three points deep into stoppage time as Toni played in Bermejo to calmly slot past Andres.