Afterwards, Georgia summoned the Russian ambassador to hand him an official note of protest over the incident, the Georgian foreign ministry said.
Russia has denied the accusation.
"We categorically deny any involvement in these events," Dmitry Peskov, a Kremlin spokesman, said.
"These declarations are not based on any reality."
Government officials said the ordnance was dropped at 8pm (1600 GMT) on Monday evening, and that bomb-disposal experts were at the scene shortly after.
They said it hit the village of Tsitelubani, about 65km west of the capital, Tbilisi.
"Fortunately it didn't explode. If it had exploded, it would have been a disaster," Shota Utiashvili, an interior ministry spokesman, said.
He said nobody was hurt.
The village is a few kilometres to the south of Georgia's breakaway South Ossetia region, a long-standing cause of friction between Russia and Tbilisi.
Russia, a financial provider for Georgia's rebellious Abkhazia and South Ossetia regions, has accused Tbilisi of pursuing anti-Russian policies.
During the rule of Eduard Shevardnadze, the ousted president, Georgia had accused Russia in 2002 of sending fighter jets on sorties over its territory, but Moscow denied any involvement.
At that time, Tbilisi alleged that Russian jets had dropped ordnance on uninhabited areas of the remote Pankisi Gorge in north-east Georgia, near the border with Russia.
Relations between Russia and Georgia deteriorated again last year when Tbilisi deported four Russian army officers, accusing them of spying.
Moscow responded by withdrawing its ambassador from Tbilisi and cutting air, sea and postal links with Georgia. Russia also deported several thousand Georgians, saying they were illegal immigrants.