After a 3-1 victory in Athens last week and reaching halftime unscathed in the return, Arsenal looked comfortable.
But when Youssef El Arabi's deflected shot left Bernd Leno wrong-footed early in the second half it gave the visitors hope and increased the tension among the home ranks.
Arsenal wasted several late chances to ease the jitters with captain Pierre-Emerick Aubameyang, restored to the starting line-up being left out of Sunday's north London derby for disciplinary reasons, guilty of a poor miss.
Olympiakos had Ousseynou Ba sent off late on and Arsenal were spared any late dramas as they avenged last season's elimination by the Greek champions to take their place in Friday's last-eight draw.
Olympiakos will take pride from a third successive away win against Arsenal but the damage was done in the first leg when they conceded two late goals having equalised.
Arsenal were comfortable until a loose ball by Dani Ceballos led to El-Arabi's goal which gave the visitors hope.
As Olympiakos pushed forward they allowed gaps at the back and Arsenal should have capitalised with substitute Martin Odegaard firing over and Nicolas Pepe having a goalbound effort deflected away while Aubameyang could have had a hat-trick.
Arsenal have now made it to the quarterfinals for the fourth time in their last five Europa League/UEFA Cup campaigns.
Speaking to BT Sport after the match, Arsenal manager Mikel Arteta admitted his side could have played better on the night.
"Really happy to be through. We had a tough opponent, so happy to be through. Sometimes we have to be fair with ourselves -- we were nowhere near the levels we could be," Arteta said.
"We were unstable with the ball and we gave it away. It's impossible when you give the ball away to have any structure. We created enough chances to win but were nowhere near the levels in which we can perform.
"We can't use any excuses. We had fresh players in the side. We started the game really well but then we had a lack of stability. We need to keep the ball and play some passes together and not just keep giving the ball away in transitions.
"When we conceded, we did get better and got more control but with their goal, we gave them some belief, which you cannot give to the opponent. We'll get better. When it comes to decision-making and execution, it's a little bit more difficult.
"We need another win in the Premier League and it's a big chance after the international break to get a run together."