Arsenal were left to rue a host of missed chances as Lea Schuller's header gave Bayern a 1-0 victory in their Champions League quarter-final first leg.
The Gunners twice saw chances cleared off the line, from Stina Blackstenius and Leah Williamson, but could not find a way to breach the hosts' defence at the Allianz Arena and were made to pay.
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Bayern top scorer Schuller nodded in her 14th goal of the season shortly before half-time and could have doubled her lead on the stroke of the interval.
Arsenal were worth at least a draw and were inches away from finding one again when Caitlin Foord rattled the Bayern woodwork from distance.
They will now need to beat Bayern in the return leg at the Emirates next week to avoid crashing out in the quarter-finals on their fourth consecutive appearance in the competition.
How Bayern held off Arsenal to take lead
In the well-attended and always imposing Allianz Arena, Arsenal seemed slightly intimidated early on but soon settled after weathering a couple of bright moments for the hosts, and they should have gone ahead when Blackstenius nodded over a free header from six yards.
The game seemed in their own hands for much of the opening period, regularly keeping Bayern pegged back inside their own half. Frida Maanum missed another strong opportunity from Blackstenius' square ball, with Maria Grohs blocking her effort from close range.
But when Bayern finally regained their composure and built up some momentum of their own, they made it tell. Schuller rose high above Laura Wienroither to nod into the top corner for the opener. She could have doubled the lead moments before the break, but lifted an effort just wide of the far post after splitting Arsenal's two centre-backs.
That momentum was punctuated by the half-time whistle and never wrestled back by the hosts. Arsenal emerged for the second period rejuvenated and almost levelled seven minutes after the interval when Foord curled a low effort past Grohs and off the far post.
With an hour gone, they went even closer when Blackstenius' powerful header beat the Bayern goalkeeper but was cleared off the line by Saki Kumagai.
It began to appear it would not be the Gunners' day soon after as they were denied by the finest of margins for a third time, when Williamson nodded goalwards from a corner and Bayern again hacked the ball clear from millimetres out.
Bayern could have given themselves a better chance yet of progressing late on with their one and only clear-cut chance of the second period, but Maximiliane Rall shot straight at Manuela Zinsberger in a rare foray forward.
Eidevall questions VAR and goal-line technology decisions
VAR reviewed an early handball call in the Bayern area after a Foord shot was blocked but saw no need to intervene, before disagreeing with Arsenal boss Jonas Eidevall's vociferous calls for a penalty after he believed Rafaelle Souza's shirt was pulled in the opposition box.
Eidevall said: "Part of me is proud, part of me is disappointed we can't be more clinical with our chances. Part of me is angry because obviously VAR was not working in the Allianz Arena tonight. It must be someone sitting there not looking at the pictures.
"Because it is clear and obvious those two penalty situations. I think that is unacceptable. There is a pull of a shirt and that handball situation in the first half. I also think that one is clear.
"When we look at this competition, tonight highlights a couple of issues. VAR is only introduced in the quarter-finals, with referees that I don't know if they have used VAR before tonight. I don't know.
"Secondly, goal-line technology is something that you can use at any stage, but it is not compulsory. Today it was not in-use because Bayern choose not to pay for it.
"Next week at the Emirates, Arsenal have chosen to use it, because we think it is fair for the competition. It is a problem for the competition that not every game has the same level of technology.
"We need to score at least one next week. But we need to still manage their threats. They are a very good counter-attacking team.
"But you need to be brave on the ball and push them into their own half. If we can continue to do that and do that in front of our own fans at the Emirates, I think we have a good chance (of progressing)."
What's next?
Arsenal face the small matter of a North London derby with Tottenham in the WSL on Saturday; kick-off 3pm.