The Real Madrid striker has not found the net in his last 11 games for Les Bleus and while three of the offensive quartet - Mathieu Valbuena, Olivier Giroud and Franck Ribery - scored goals, Benzema once again proved too clumsy.
He was even booed by the Stade de France crowd, who are not known for their patience and understanding.
"Strikers need confidence, and they do not need to be booed by the crowd," midfielder Blaise Matuidi told reporters.
Benzema has been repeating that he is serene, yet the last of his 15 France goals from 56 caps came in June when he scored the third in a 4-0 friendly win against Estonia in Le Mans.
"He had chances. I would have been worried if he had not had them," coach Didier Deschamps told reporters after Benzema missed several clear chances against Georgia on a bumpy Stade de France pitch.
"He would like to be more decisive. I've left him on the field until the end because he had not played 90 minutes for a long time. But I would have preferred that he scores this goal."
France, however, are top of Group I with 10 points from four games and lead Spain by two points after the world and European champions, who visit the Stade de France on Tuesday, only managed a draw against Finland in Gijon.
Benzema's team mates said they are not concerned by the 25-year-old's failings.
"I'm not worried about Karim, we must give him a bit more time," Ribery told reporters.
"He is a great player with a lot of qualities, unfortunately he has not scored (against Georgia) but I hope it will change on Tuesday.
"I also went through difficult periods, but he works hard."
Benzema has scored 15 goals from 35 appearances in all competitions with Real this season - compared to 31 from 50 games last season.
Deschamps paired him with Giroud on Friday in an attempt to pressure the Georgia defence, but also because he felt that Benzema needed a hand.
"With Karim, we try to find each other but it's still perfectible," Giroud told reporters.
Benzema's place in the starting line up for Tuesday's clash with Spain is now at risk as Deschamps will probably revert to a more secure formation with a lone striker.
Giroud, who scored the late equaliser six minutes after coming on as a substitute for the Real player in the 1-1 draw when the teams met at the Vicente Calderon in October, is likely to be preferred.