"Two drones belonging to the Israeli enemy violated Lebanese airspace (at dawn)... over the southern suburbs of Beirut. The first fell while the second exploded in the air causing material damage," an army statement said.
The early morning incident came hours after Israel launched airstrikes in neighbouring Syria.
The army said it had sealed off the area of Beirut and that the military police had opened an investigation.
Earlier, a Hizballah spokesman, Mohamed Afif, said one of the two drones was rigged with explosives.
"The first drone fell without causing damage while the second one was laden with explosives and exploded causing huge damage to the media centre," Afif told the National News Agency.
"Hezbollah did not shoot down any drone," he said.
"The first drone did not explode and it is now in the possession of Hezbollah which is analysing it."
He expected Hizballah chief Hassan Nasrallah to give a "staunch response" during a speech later in the day at a pre-scheduled group event.
Iran-backed Hizballah - considered a terrorist organisation by Israel and the United States - has fought several conflicts with Israel.
The last confrontation was a 33-day war in 2006 that killed 1,200 in Lebanon and 160 in Israel.
Hostilities have spilled over into Syria, where Hizballah - like Israel's arch-foe Iran - is fighting on the side of the government of Bashar al-Assad.
On Saturday night Israeli jets carried out strikes in Syria to prevent an Iranian force from launching an attack on Israel with drones armed with explosives, Israel's army said.
Lebanon and Israel are technically still at war, and Beirut regularly accuses Israel of violating its airspace with planes and drones.
Questioned by AFP, the Israeli army declined to comment on the Lebanese army's claim about the drones.
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