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.
"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."
"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.
