The 2009 champions, who had lost their first two games, needed a late header from a corner by central defender Ciani to win a lively contest at Parc des Princes.
Bordeaux opened the scoring against the run of play with a headed goal by France midfielder Alou Diarra, also from a corner, on 67 minutes.
PSG, who have struggled in the past few years but started the campaign in convincing fashion, had levelled courtesy of a goal from close range by forward Guillaume Hoarau 14 minutes from time.
New Girondins coach Jean Tigana left Yoann Gourcuff out of his starting lineup, suggesting the France playmaker could soon be leaving the club.
Gourcuff, who has asked for a move according to French media, came on for the last 16 minutes and kicked the corner from which Ciani headed the winner.
Bordeaux suffered at times against PSG but looked much stronger than in their first two outings, partly because goalkeeper Cedric Carrasso, who made a few spectacular saves, and Ciani were back from suspensions.
PARIS FRUSTRATED
PSG players felt frustrated as they had looked to be the more dangerous side.
"We offered them presents with defensive errors and also by being clumsy up front," Hoarau told reporters. "If you do those things, you can't expect to win. We have plenty to work on."
Bordeaux, by contrast, breathed a sigh of relief.
"We needed that," Carrasso told reporters. "Everybody fought for it and that was the key."
Promoted Caen dropped down a gear after a stunning start when they were held to a goalless draw at Montpellier earlier on Sunday.
The side from Normandy, who had started their campaign in style with wins over Olympique Marseille and Olympique Lyon, showed little initiative in a game played in stifling heat and the best chances went to Montpellier.
Caen stay second but fall two points behind surprise early leaders Toulouse, the only team on maximum points after three games.
PSG drop to seventh, five points off the pace, while Bordeaux move up to 17th in the 20-strong league.
