He warned that Israelis in the northern settlements, which are close to Lebanese borders, will not be safe. 
From now on, "we will confront the drones in the skies of Lebanon and will bring them down," said the head of the Iran-backed movement.
On Sunday, two Israeli drones went down over Beirut's southern suburbs: One fell near the movement's press office, and the other exploded in the air.
The Lebanese army said in a statement that two Israeli "enemy" reconnaissance planes breached Lebanon's airspace over the area of Mouwad-Hey Madi in Beirut's southern suburbs at dawn Sunday.
One drone fell and the second exploded in the air, causing only material damage on the ground, the army added.
A Hezbollah official said the group had not shot down the two drones.
Nasrallah said the dawn attack is the first Israeli aggression in Lebanon since August 14, 2006.
"Lebanon will face a very dangerous situation if this incident goes unaddressed," the Hezbollah chief said.
As is often the case, the Israeli army said it does not comment on foreign reports.
A Hezbollah official, who requested anonymity, told dpa that the group had collected the parts of one drone that had crashed near the group's press office in the suburb of Mouwad.
The area is a stronghold of the movement, which is mainly financed and trained by Iran.
Lebanese President Michel Aoun said the "Israeli aggression" in Beirut's southern suburbs is a "chapter of continuing violations of Security Council Resolution 1701."
That resolution called for a ceasefire that ended a 33-day war between Lebanon and Israel in 2006.
Lebanese Prime Minister Saad Hariri called the alleged Israeli drones an "open aggression on Lebanon's sovereignty."
The incident comes as Israel is engaged in drone activities in the skies above Lebanon's war-torn neighbour, Syria.
Iran and Hezbollah are staunch supporters of Syrian President Bashar al-Assad in his fighting against an armed rebellion that erupted in 2011.
Late Saturday, the Israeli army said that it had thwarted an Iranian attempt to attack Israeli targets using "killer drones" and that it had struck several targets in Aqraba, south-east of Damascus, in Syria.
"The strike targeted Iranian Quds Force operatives and Shiite militias, which were preparing to advance attack plans targeting sites in Israel from within Syria over the last number of days," the army said in a statement.
Army spokesman Jonathan Conricus said Sunday that the operatives' intention had been to fly a number of attack drones towards targets in northern Israel, with each drone carrying several kilograms of explosives.
Senior commanders of Iran's Islamic Revolutionary Guard Corps' (IRGC) Quds Force, the elite fighting force that carries out special missions abroad, were behind the attempt, Conricus said.
Nasrallah denied Sunday that the Israeli attack targeted Quds Force.
"Israel did not attack a Quds Force position in Syria but rather a house containing Hezbollah fighters," Nasrallah said.
"If Israel kills any of our members in Syria... we will respond from Lebanon... we tell Israelis on the border to beware," Nasrallah said.
Five fighters loyal to the Syrian government were killed in the overnight Israeli attack near Damascus, a monitoring group said.
Those killed were two Hezbollah fighters, one Iranian as well as two others whose identities have not been established yet, the Syrian Observatory for Human Rights added.
"None of those killed were civilians. All were fighters," observatory head Rami Abdel-Rahman told dpa.
A Hezbollah spokesperson confirmed that two members of the group were killed in the Israeli attack.
Israeli Prime Minister Benjamin Netanyahu held a security assessment in northern Israel on Sunday, according to a statement. 
"We will not tolerate aggression against Israel from any country in the region. Any country that allows its territory to be used for aggression against Israel will face the consequences," he said.
US Secretary of State Mike Pompeo called Netanyahu and expressed support for Israel's right to defend itself from threats posed by the IRGC, according to State Department spokesperson Morgan Ortagus.
They also discussed how Iran is leveraging its foothold in Syria to threaten Israel and its neighbours, Ortagus added in a statement.
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