Arsenal went into the reverse fixture with a 1-0 lead from the first-leg but it was Olympiakos who went through courtesy of a Youssef El Arabi goal in the 120th minute.
After a goalless first half Olympiakos broke the deadlock in the 53rd minute through Pape Abou.
With aggregate score at 1-1, the game went into extra-time before Pierre-Emerick Aubameyang scored with a superb acrobatic kick to put Arsenal 2-1 on aggregate.
They failed to hold on as El Arabi struck to bring score to 2-2 and put the Greek side ahead on the away goals rule.
With seconds left in the game, Aubameyang had a golden chance after a poor clearance fell kindly to him but he shot wide with the keeper to beat.
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