The Gunners’ 3-1 first-leg win away from home meant they were huge favourites to reach the last eight of the competition, and that result proved vital as Mikel Arteta’s side were well below their best at the Emirates.
Youssef El Arabi should have scored in the first half after racing onto a routine goal kick in behind the Arsenal defence, but Bernd Leno made a top save to deny him.
Pierre-Emerick Aubameyang then wasted a glorious chance at the other end as the first half ended goalless, but Olympiacos were ahead soon after the restart.
El Arabi made amends for his earlier miss as he found the back of the net with a deflected effort after 51 minutes.
Olympiacos tried to find the two extra goals they needed on the night, but lacked the quality to unsettle Arsenal after that – and Ousseynou Ba’s late red card for dissent made things slightly more comfortable for the Gunners.
Arteta will be far from happy with the result on the night, but Arsenal are in the hat for Friday’s quarter-final draw.