Marseille's two late goals in a 3-2 victory at Borussia Dortmund in the other final Group F game meant Didier Deschamps' team qualified for February's first knockout round and Olympiakos had to settle for third place and the Europa League.
The Greeks ended the group stage on nine points, just one behind the French. Arsenal had secured top spot and progression before Tuesday's game.
"We're out of the Champions League but we're proud of the way we played today," said Olympiakos coach Ernesto Valverde.
The Spaniard, as well as visiting manager Arsene Wenger, identified Olympiakos' first group game which ended in a 1-0 home defeat by Marseille as the main reason the Greeks failed to clinch a rare qualification to the knockout stage.
"Our first defeat at home to Marseille cost us a lot," said Valverde. "Football is about doing the right things at the right time.
"Despite losing our first two games, we managed to push for qualification right up to the last game, so that's a success for us. Now we'll see how far we can go in the Europa League."
"I thought overall Olympiakos deserved to win the game but, unfortunately for them, it's not enough because they will not qualify for the Champions League. Marseille made a super result," Arsenal manager Wenger told reporters.
"They missed the first game at home against Marseille and it's very difficult to get that back. Olympiakos deserved a better place. They paid a heavy price for not starting the championship before their first game.
"Offensively and defensively, we were not convincing tonight. It was a fantastic experience [for the
