The Gunners, who are now in the semi-finals for the first time since 2013, beat the German side 2-0 on the night and thus going through 2-1 on aggregate.
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Maanum's opener levelled the scores on aggregate, but it was a goal worthy of winning any European tie.
Some lovely one-touch football from Arsenal left the Bayern defence spellbound in scenes that wouldn't have looked out of place during the men's teams peak years under Arsene Wenger.
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Captain Leah Williamson teed up Maanum with a cheeky flick before the Norwegian brilliantly chipped the ball over Bayern goalkeeper Maria Luisa Grohs and into the net.
The strike had manager Jonas Eidevall on his feet at a rain-soaked Emirates Stadium while the vast majority of the 21,307 in attendance - a record crowd for an Arsenal Women's Champions League match - went wild.
That set the 15-time English champions on their way to a famous win with Sweden's Stina Blackstenius scoring the winner in the tie on 26 minutes.
Arsenal, who could still win a Champions League, Conti Cup and Women's Super League treble this season, will face the winners of the Paris Saint-Germain vs Wolfsburg tie in the last four.
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