An 83rd minute header from Michael Ciani condemned Maccabi Haifa to their second consecutive Champions League defeat.
The Israel frustrated the French champions Bordeaux, restricting them to half chances and looked threatening on the counter-attack at the Stade Chaban Delmas.
Haifa finished with 10 men after Baram Kayal was shown a straight red card for a dangerous challenge two minutes from time.
Haifa manager Elisha Levy said: "We played the right way and they hardly created scoring opportunities. This is frustrating. I felt we could have won the match, but we lost and now we need to make sure we remember all the positives from tonight."
Captain Yaniv Katan said: "This is very disappointing because we played so well. This is cruel, but we need to know how to take the good things from this match and look forward to our next encounter."
Having lost their opening group match against Bayern Munich, Haifa remain pointless ahead of a double-header against Juventus, with the first match in Turin on October 21.