Centre back Ciani headed home from a corner seven minutes from time to rescue a disappointing display by the French champions.
Substitute Lior Refaelov had the best chance for the visitors in the 79th minute but he was denied by keeper Cedric Carasso. Maccabi Haifa ended the match with 10 men after Baram Kayal was shown a straight red card in the 88th minute for a dangerous tackle on Benoit Tremoulinas.
Bordeaux extended their unbeaten run in all competitions to 21 matches but they failed to impress at their Chaban-Delmas stadium.
Haifa were the first to find their range when a powerful shot from Eyal Meshumar from just outside the box was parried away by Carasso.
Carasso tipped away a Mohammad Ghadir attempt shortly before the half hour as the hosts struggled to gain a foothold in the match.
Yoann Gourcuff was replaced by Fernando at halftime after the France playmaker suffered a thigh injury.
"He has a muscle strain behind his left thigh," Bordeaux coach Laurent Blanc told reporters.
"He will take some time off, it will do him good. Bordeaux will be penalized and maybe the French team will be. We have to wait for a medical diagnosis."
France play the Faroe Islands in a 2010 World Cup qualifier in Guingamp on October 10.
The home side produced a better second-half display and after substitute David Bellion wasted two good chances Ciani found the back of the net with a fine header from a Gregory Sertic corner.
Bordeaux have four points from two matches, the same as Bayern Munich who sit top on goal difference.
