Direct flights have been suspended since the two nations went to war over the Russian-backed separatist Georgian region of South Ossetia last year, only four months after they had resumed following an 18-month embargo.
"It's great that there are flights because we have many friends and relatives in Russia and we couldn't see them without flying through other countries," said Nana Samadalashvili, a Georgian passenger.
"We don't want to play politics, we just don't want to have to go through all of this."
Nino Giorgobiani, a spokeswoman fro Georgian Airways, said the airline was negotiating more flights on January 16, 20 and 24.
The company had planned flights on December 29 and 30 but cancelled them, saying Russian permission had come too late.
Last month, Dmitry Medvedev, the Russian president, said that he saw no obstacle to resuming flights, granting visa-free travel to Russia for Georgian citizens and lifting an import ban on the country's wine.
But Moscow has ruled out talks with Mikheil Saakashvili, the Georgian president.
In the first sign of a thaw in relations, Georgia and Russia announced on December 24 that they had agreed to re-open their land border to traffic.
Under Swiss mediation, they agreed to open the land crossing at Upper Lars, the only one that does not pass through either South Ossetia or Georgia's other separatist region of Abkhazia.
Fighting between the two countries erupted in August 2008, when Russian forces poured into Georgia to repel a Georgian military attempt to retake South Ossetia.
Russian forces later mostly withdrew to within South Ossetia and Abkhazia, both of which Moscow recognised as independent states, a move so far followed by only Nicaragua, Venezuela and the Pacific island state of Nauru.