The Gunners' victory on the night was not enough to stop them exiting the competition with an away-goals defeat following a 3-3 aggregate scoreline.
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Bayern enjoyed much of the possession in the first half following the goal, but Luiz Gustanvo smashed his effort over the bar before Toni Kross was denied by Lukas Fabianski.
The home team started the second half strongly and they nearly extended their aggregate advantage on 49 minutes when Kroos whistled a shot narrowly wide.
Bayern continued to press as they looked to kill the tie off, but Gustavo could only curl the ball the wrong side of the post before Fabianski denied Arjen Robben with the winger through on goal.
Arsenal then threw everyone forward in an attempt to score the goal that would see them into the quarter-finals, but they were unable to breach the Bayern defence for a third time.