The 1998 world champions failed to make the breakthrough and look increasingly likely to have to negotiate a knockout match if they are to reach Brazil
Although the visitors dominated possession, it was the hosts who went closest to winning the game as they hit the post in the second half. In stoppage-time Olivier Giroud had a fine headed chance but saw his effort saved.
Worryingly for Deschamps, it was the fifth consecutive game in which his side has failed to score, while it has now won just once in its last seven matches.
Temuri Ketsbaia's men started positively, with Jano Ananidze surging past Josuha Guilavogui before Moussa Sissoko halted his progress.
France responded well and almost took the lead with 12 minutes gone as Karim Benzema opened up the space with a smart run before teeing up Franck Ribery, who darted into the box, cut inside and flashed a shot narrowly wide.
Jaba Kankava connected well with an ambitious volley after 17 minutes that swerved wide as the Georgians continued to show their attacking intent.
Hugo Lloris had to be alert to keep out a Tornike Okriashvili effort from the edge of the box eight minutes later.
Opposite number Giorgi Loria should have been called into action at the other end in the latter stages of the half, but Olympqiue de Marseille winger Mathieu Valbuena rifled his shot over the bar from inside the box.
After a strong start to the second half, France was nearly hit on the counterattack but the home side was denied the opening goal in dramatic fashion.
Lloris fumbled Okriashvili's initial shot and the midfielder, following up for the rebound, saw his second attempt crash off the post before the visitors scrambled it to safety.
That close call spurred the French on and Giroud was inches away from getting on the end of Ribery's cross, only for Aleksandr Amisulashvili to make a crucial intervention.
Giroud was guilty of spurning a glorious opportunity in the 70th minute, receiving the ball fortuitously after Patrice Evra's wayward shot fell to him only to flash his shot over the bar.
There was still time for Loria to deny Valbuena and Ribery with two excellent late stops, while Laurent Koscielny headed wide from close range as the French drew a blank once again.
Giroud took a tumble in the box deep into stoppage-time but his desperate appeals for a penalty were waved away.