France's scoreless run was extended to five games as Georgia dented their hopes of beating Spain to top spot in World Cup qualifying Group I.
Didier Deschamps' side were well below their best in the first half and were almost punished when Tornike Okriashvili hit the woodwork early in the second.
They stepped up a gear after that but could not find a way past home goalkeeper Giorgi Loria, who crowned a good display with two stunning saves in stoppage time to earn his side a point.
France went into the game having failed to score in their last qualifier, a 1-0 loss to Spain, as well as friendly defeats to Brazil and Uruguay and a goalless draw with Belgium.
Goalkeeper and captain Hugo Lloris had warned his colleagues at the other end of the field that "that will have to change if we want to win games'', but another stalemate ensued in Tbilisi.
Deschamps opted to partner Arsenal's in-form hitman Olivier Giroud with Karim Benzema, but neither they nor substitute Andre-Pierre Gignac were able to find a breakthrough.
Neither team had a sight of goal in the opening 10 minutes and when France did create a chance, Franck Ribery shot wide. Georgia's Jaba Kankava took it upon himself to break the shackles, but his 30-yard volley flew wide.
Okriashvili forced Lloris into the first save of the night after Laurent Koscielny only half-cleared Alexander Kobakhidze's free-kick, and Benzema was then off target from a similar situation at the other end.
Loria parried another effort from the Real Madrid man and Mathieu Valbuena shot over as France were held at bay until half-time. And things so nearly got worse for Les Bleus just four minutes into the second
period when Okriashvili drew a sharp save from Lloris and fired a follow-up effort against a post.
Lloris then fielded Nikoloz Gelashvili's cross-shot, while Benzema failed to end an international drought stretching back to last June's brace against Estonia when he headed Bacary Sagna's cross over the bar.
It was his last contribution before being replaced by recalled Marseille striker Gignac, who soon also failed to steer a Sagna cross on target.
Giroud's header was saved and both he and full-back Patrice Evra missed the target from distance as France took control but failed to make it count.
Loria parried Ribery's shot and then saved from Valbuena, and Koscielny headed wide. The pressure continued to mount on Georgia's goal and Loria was booked for time-wasting before producing the decisive contribution at the last for Temuri Ketsbaia's underdogs.
First Giroud's header was tipped around the post and, from the resulting corner, Loria produced more heroics to deny Sagna and ensure France were shut out again.
The result, coupled with Spain's win in Finland, leaves France three points off the pace and urgently needing a win in Belarus on Tuesday to keep up the
pressure on the reigning world and European champions.