Bayern had two late chances to take the three points, but Spanish substitute Edna Imade failed to hit the target both times.
The hosts reached the semi-finals of last season's Champions League and were playing a first competitive match at the Sportpark Unterhaching stadium, which is designated to be their long-term home.
A comfortable victory looked to be coming their way after a dominant opening 40 minutes.
As well as the goals that came their way, Bayern missed numerous opportunities, with Klara Buhl firing wide and former Chelsea attacker Pernille Harder also wasting an early attempt.
After Gwinn's opener, Buhl had an effort saved by Yamashita, and Jovana Damnjanovic shot over when well placed.
A deflected cross from City's Lauren Hemp bounced off the top of the crossbar and Shaw headed over from the following corner, before Casparij's own goal gave City an uphill task.
Bayern's Japanese midfielder Momoko Tanikawa could not convert two opportunities to make it 3-0 and from then on City were the better side.
Mead gave the visitors hope and even Shaw's missed penalty did not dampen spirits, with Wamser securing a hard-earned point for City.
Just like last year, the league phase of the Women's Champions League features 18 teams in one division, with each side playing six different opponents, three at home and three away.
Bayern play next in Portugal against Benfica on 30 September, and Manchester City have home advantage against Real Madrid on 1 October.