"We downed another [Georgian] drone over the village of Gudava in the Ochamchir district at about 17.10 Moscow time (13.10 GMT)," Merab Kishmaria said.
However, Georgia's President Mikheil Saakashvili immediately denied the report.
"I have just spoken with the [Georgian] defense minister," Saakashvili told reporters. "Nothing has been downed."
According to information from Abkhazia, the republican Air Defense has shot down four Georgian surveillance drones this year, including one on March 18, April 20 and a further two on May 4.
The Georgian Foreign Ministry called Abkhazia's claims "absurd," and said they were aimed at escalating tensions in the region.
However, Georgia accused Russia of shooting down an unmanned reconnaissance plane on April 20 - a claim Russia flatly denied, calling video footage provided by Georgia a fake.
Relations between Moscow and Tbilisi have drastically deteriorated since the Kremlin called for closer ties between Moscow and the two Georgian breakaway republics of Abkhazia and South Ossetia in mid-April.
Moscow has increased the number of Russian peacekeepers in Abkhazia to around 3,000 from 2,000, but said the rise was within the limits of agreements on troop numbers signed by the Georgian leadership.
The Russian Foreign Ministry said Georgia was to blame for fueling tensions in the conflict-stricken region by conducting reconnaissance flights over Abkhazian territory.
Abkhazia, alongside another Georgian breakaway republic, South Ossetia, broke away from Georgia in the early 1990s following the collapse of the Soviet Union. Between 10,000 and 30,000 people were killed in the Georgian-Abkhazian conflict and some 3,000 in Georgian-South Ossetian hostilities. Georgia is looking to regain control over the two de facto independent republics.
