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The Gunners were a goal up after the first leg of the tie, having won 1-0 away, the previous week.
However, with a 2-1 lead over Arsenal and an away goal's advantage, Aubamayang had the opportunity to prevent Olympiacos' progression to the next round.
With just seconds remaining, the Gabon international missed from about six yards out.
When asked after the match what had gone wrong for him, he replied: "I don't even know.
"It can happen but I do not know how I missed this chance. I was tired, I had some cramps but it is not an excuse."
Speaking after the match, Gunners boss Mikel Arteta said: "It hurts a lot."
However, he would not criticise his players, of whom he said he was "proud."
He said: "It is cruel. It is a massive disappointment.
"It would have been an incredible lift if we had managed to go through the tie. The players have my backing for the way they played but it is painful."
Stage 15, Tolmezzo - Sappada (176km)