Bayern had playmaker Thomas Mueller sent off for a second booking and five minutes from the break defender Marc Planus scored the winner for the French champions from close range.
Bordeaux then missed two penalties and Bayern had another player sent off in an eventful but goalless second half.
The win put Bordeaux top of Group A with seven points from three matches. Juventus follow on five points after beating Maccabi Haifa 1-0 and Bayern dropped to third place on four points.
"We made sure it remained tense until the end," Bordeaux coach Laurent Blanc told reporters.
"Bayern put us under pressure at first. They were a bit lucky to score but that actually helped us get into the match.
"We reacted well by scoring two goals but should of scored a few more," he added. "Missing two penalties is not acceptable. That's something we'll have to work on in training. Set pieces are important."
Four-times European champions Bayern, missing three leading players in Franck Ribery, Arjen Robben and Ivica Olic, fought bravely with 10 men and narrowly missed out on an equaliser when Italian forward Luca Toni hit the post with a header on 72 minutes.
"We were actually better in the second half with 10 men than in the first half with 11," Bayern coach Louis van Gaal told reporters.
"We made far too many mistakes in the first half after opening the scoring and that's irritating," he added.
French champions Bordeaux, who have lost their last two Ligue 1 matches, wasted many chances to kill off the match, notably when playmaker Yoann Gourcuff missed their first penalty on 66 minutes.
Their second came two minutes from time after defender Daniel Van Buyten was red-carded for bringing down Morocco striker Marouane Chamakh. Substitute forward Jussie missed from the spot.
