Mikel Arteta's side headed to the Allianz Arena for their second-leg tie against the Bavarians following a 2-2 draw at the Emirates Stadium last week.
However, Joshua Kimmich's 63rd-minute header proved decisive, securing Bayern's place in the semi-finals of Europe's premier competition.
Despite their efforts, the Gunners were unable to breach Manuel Neuer's goal on the night. This setback came shortly after their Premier League title hopes were dealt a blow, having suffered defeat to Aston Villa last weekend.
RB Salzburg qualify for Club World Cup at the expense of ArsenalArsenal's loss in Bavaria resulted in Red Bull Salzburg securing the final European spot in the Club World Cup competition, which is scheduled to take place in the United States during the summer of 2025.
The inclusion of the Austrian side may come as a surprise, but according to FIFA regulations, nations are allowed only two entries unless a country has multiple continental champions. As a result, Salzburg qualifies for the competition despite being ranked below Liverpool, Barcelona and AC Milan.
Europe received 12 spots for the expanded format of the competition which will be played in June and July next year, and the event will feature a total of 32 teams representing six confederations.
The Gunners would have needed to win this season's Champions League to secure a guaranteed spot in the competition in the US.
Messi and Ronaldo set to miss 2025 Club World CupBarcelona, Manchester United, Liverpool, AC Milan, Lionel Messi's Inter
Read on sportsmole.co.uk