Bayern 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.
"It was a game of two halves, I think. (The first half) was probably not good enough for our standards," Mead told UEFA.
"But again we showed what the team is about, showed our character."
Arsenal, the 2025 Champions League winners, made a faltering start to their bid to recover the continent's premier club title.
Renee Slegers's side needed an injury-time penalty from Spain international Caldentey to nab a 1-0 win at home to Danes HB Koge.
The Gunners looked certain to be heading for a goalless stalemate until new signing Selina Cerci went down in the box under Yukina Sakabe's 92nd-minute challenge.
A Sakina Karchaoui penalty inside the final 20 minutes ensured Paris Saint-Germain left the Estadio Alfredo Di Stefano with a 1-1 draw after Felicia Schroeder's second-minute opener for Real Madrid.
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.
A brace either side of half-time by Marit Bratberg Lund gave Benfica a 2-1 win at Juventus, despite the hosts scoring late through Alba Redondo.
