Georgian authorities detained the officers on Wednesday, prompting angry statements from the Russian defence minister who denounced Georgia as a "bandit" state.
Bilateral ties have long been strained over Georgia's bid to join NATO and Moscow's close ties to Georgia's breakaway provinces.
Russia's Foreign Ministry said that Moscow decided to launch a "partial evacuation of Russian personnel in Georgia and their family members in connection with a growing threat to their security."
It said that the evacuation will start tomorrow and will be conducted by the Emergency Situations Ministry planes.
The ministry also alerted all Russians to refrain from trips to
Georgia.
Georgia's President Mikhail Saakashvili denounced the Russian move as "hysteria".
"Russian personnel and their families face absolutely no threat here," he said.
Mikhail Svirin, a spokesman for the Russian Embassy in Georgia, said that Ambassador Vyacheslav Kovalenko, some of the staffers and all diplomats' families would leave Georgia tomorrow.
Earlier in the day, the Russian Embassy in Georgia indefinitely suspended the issuance of visas to all Georgian citizens.
Relations between Moscow and Tbilisi have become increasingly tense after the pro-Western Mr Saakashvili came to power following Georgia's 2003 Rose Revolution.
Georgian officials have accused Russia of backing separatists in
Georgia's breakaway provinces of Abkhazia and South Ossetia and making efforts to undermine Mr Saakashvili's government - allegations Russia has denied.
Russian Defence Minister Sergei Ivanov called the charges against the officers absurd and demanded their immediate release.
"All of this is aimed at provoking the situation and raising the degree of escalation to the maximum level in order to deflect attention from problems that exist in Georgia," Mr Ivanov said.
"Russia will react appropriately and responsibly."
Officiers detained
Interior Minister Vano Merabishvili said four Russian military intelligence officers were detained in the Georgian capital and the Black Sea port of Batumi on charges of espionage.
A fifth officer was detained later that day, the ministry said in a statement.
Georgian security forces maintained a presence around the Russian military headquarters in Tbilisi for a second day on Thursday demanding the handover of another Russian officer accused of spying.
Russia maintains two military bases in Georgia as Soviet-era holdovers. One is to be closed in 2007 and the other a year later.
Russian peacekeepers also have been deployed to Georgia's separatist provinces of Abkhazia and South Ossetia that have enjoyed de facto independence after breaking away in the early 1990s.
Russia has also granted most residents of the two regions citizenship.
Mr Saakashvili has vowed to take Abkhazia and South Ossetia back into the fold, and he accused the Kremlin last week of "gangster occupation" of the two provinces in a speech before the United Nations.
Mr Ivanov alleged that the detentions were part of Georgia's efforts to force Russia to withdraw its peacekeepers.
"All this is done to squeeze our peacekeepers and to make their status illegitimate using any means, thus violating fragrantly all previous agreements," he said.