The hosts raced into a 2-0 lead thanks to Giulia Gwinn's early strike and a Kerstin Casparij own goal inside the first 30 minutes.
Beth Mead halved the deficit for the Women's Super League champions on the stroke of half-time.
City looked set to begin the league phase with a loss, as Khadija Shaw was denied from the spot by Bayern goalkeeper Ena Mahmutovic in the second period.
But German forward Carlotta Wamser popped up with four minutes remaining to ensure the spoils were shared in Bavaria.
In the other early kick-off, Inter Milan got the better of Swedish side Hacken 1-0 - Haley Bugeja's fifth-minute goal was enough to give the home side three points.