Lea Schuller's header on the 39-minute mark proved to be the difference, as the Gunners failed to make their second-half dominance count during a night of controversial officiating.
The German powerhouses unsurprisingly enjoyed the lion's share of possession early doors, but neither goalkeeper was worked too hard, as Manuela Zinsberger watched efforts from Schuller and Georgia Stanway fly wide.
Zinsberger was called into action to beat away Schuller's 31st-minute strike, but the Bayern forward was not to be denied eight minutes later, rising above Laura Wienroither to head home from Maximiliane Rall's cross.
Schuller was inches away from bagging her second before the break - firing just wide of the post - but Jonas Eidevall's side were a team reborn in the second period, as Caitlin Foord struck the post herself with 51 minutes gone.
Bayern were indebted to a stunning goal-line clearance in the 61st minute, as Saki Kumagai hooked Stina Blackstenius's header away from danger, although Arsenal were aggrieved when an apparent tug on the shirt of Rafaelle was not given.
The red and white shirts kept on coming, though, and Schuller went from hero in the final third to hero in her own penalty area, clearing Leah Williamson's strike off the line in the 74th minute before Arsenal were up in arms again.
Bayern goalkeeper Maria Luisa Grohs appeared to pick up a clear backpass from Glodis Perla Viggosdottir, but referee Ivana Projkovska inexplicably did not take any action.
Arsenal's efforts were ultimately in vain as they prepare to host Bayern at the Emirates in the second leg next Wednesday, with either Wolfsburg or Paris Saint-Germain awaiting the victors.