The Gunners were beaten 1-0 on Thursday at the Emirates Stadium but the result could not prevent Mikel Arteta's side prevailing 3-2 on aggregate.
After a tight first half, Youssef El-Arabi, who also scored in the first leg, weaved through the box to score and give the Greeks hope after 51 minutes.
But Arsenal resisted further damage and the visitors' hopes were ended when Ousseynou Ba was sent off for a second bookable offence seven minutes from time.
Spanish striker Borja Mayoral scored both goals for the Italian side either side of a reply from Junior Moraes.
An own goal from Jesus Vallejo and a last-minute penalty from Eirik Hestad gave the Norwegians a 2-1 success but Granada, who scored a crucial away goal through substitute Roberto Soldado, had enough for a 3-2 aggregate triumph.