"At 14:16 Moscow time today [10:46 GMT], another unmanned aerial vehicle was shot down, in the Ochamchiry area. We are currently looking for the wreckage," Merab Kishmaria said.
He said the second plane was brought down at 15:13 Moscow time [11:13 GMT], and that UN mission observers were present when the drone was shot down.
Georgia's Interior Ministry has denied the reports.
"This is all lies. Our drones made no flights over Abkhazia today," a ministry spokesman said.
Georgia has so far recognized the destruction of only one unmanned plane, categorically denying all other incidents.
Abkhazia earlier claimed its air defenses had shot down five Georgian surveillance drones this year, one on May 8, two on March 18 and April 20, and a further two on May 4.
On April 20, Georgia accused Russia of shooting down a drone over Abkhazia, a claim Russia flatly denied, calling the 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.
Russia's Foreign Ministry has accused Georgia for fueling tensions in the conflict-stricken region by conducting reconnaissance flights over Abkhazian territory.
Abkhazia and 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.
Abkhazia has said fragments of all downed planes identified them as Israeli-made Hermes 450s.
Georgian President Mikheil Saakashvili earlier said Georgia had bought a total of 40 drones.
