Elias Bou Saab displayed a pair of drones - one intact and the remains of the other - and used slides to give a detailed account of the alleged Israeli operation.
He said the devices were "advanced military production" and even listed the name and address of one Israeli company he said made some of the components.
"For the first time we see drones loaded with explosives fly over Beirut's airport and threaten the security of civil aviation," he added.
The 25 August drone incident in Beirut, which came a day after Israeli strikes targeted Hezbollah operatives in Syria, drastically raised tensions in the region.
Lebanon's government and Hezbollah chief Hassan Nasrallah both described the apparently botched operation as an act of aggression.
Israel's and Hezbollah's respective patrons, the US and Iran, have been locked in an escalating tit-for-tat proxy war in recent months.
Bou Saab said that the data retrieved from the intact drone - from which the explosive charge was purportedly removed for the press conference - offered a set of detailed times and locations.
The operation was initiated 11.6 kilometres (7.2 miles) off the coastal airstrip of Habonim, south of the Israeli city of Haifa, he said, adding that the explosives-laden device had been guided by another larger drone in the area.
To reach its presumed target, that device flew over the Jnah district near Beirut International Airport, in what the defence minister said would have posed a risk to civilian air traffic.
Following the Beirut incident, Hezbollah had vowed to take down Israeli drones overflying Lebanon and on 9 September claimed it had downed and seized one.
The Israeli army confirmed that one of its devices "fell" in Lebanon but it has not commented on the 25 August incident.
Agencies contributed to this report.
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