Russian forces moved into Georgia in support of South Ossetian separatist forces last month after Tbilisi launched an offensive to retake the region.
The Georgian interior ministry said the drone was shot down on Monday morning near the town of Gori, about 30km from the de facto border with South Ossetia
"It was flying over the territory between the villages of Khurvaleti and Tsitelubani," Shota Utiashvili, an interior ministry spokesman, said. "We believe it was patrolling the territory where the Baku-Supsa [oil] pipeline runs."
The ministry displayed what it said was the downed Russian drone during a news conference. The aircraft was around one metre in length and 1.5 metres wide.
Utiashvili said it was a short-range drone capable of taking photographs, and suggested it had been launched from Russian positions at a "security zone" just a few kilometres north of Gori.
Russian forces continue to hold positions inside undisputed Georgian territory. A French-brokered deal which requires Russian forces to withdraw from "security zones" adjacent to South Ossetia and another breakaway region, Abkhazia, by October 10.
A European Union mission of at least 200 monitors begins deploying this week to observe the pullback.
Before last month's war, Georgia flew unmanned reconnaissance aircraft over Abkhazia. A United Nations report concluded one of them had been shot down by a Russian jet, though Moscow denied this.
One Georgian police officer was killed and three others were wounded on Sunday by fire from separatists fighters in Abkhazia. Another two Georgian officers were wounded on Monday when they went to the area to investigate and stepped on a landmine.