"What happened yesterday night was a suicide drone attack on a target in Beirut's southern suburb," Hassan Nasrallah said in a speech broadcast to thousands of supporters.
He said it was the first such "hostile action" since the 2006 war between Hezbollah and Israel and violated the UN Security Council resolution that ended the conflict.
"We will do everything to prevent" such attacks, he said.
"The time when Israeli aircraft come and bombard parts of Lebanon is over."
"I say to the Israeli army along the border: from tonight be ready and wait for us," he went on.
"What happened yesterday will not pass."
Addressing Israelis, he said: "Do not live, do not rest, do not be reassured and do not bet for a single moment that Hezbollah will allow... aggression of this kind."
Nasrallah was speaking after the army said two Israeli drones had violated the airspace over Beirut.
Israel has not commented.
Hezbollah said earlier that one of the drones had damaged a media centre belonging to the group in a residential building.
The incident came hours after Israeli strikes overnight in Syria, which Israel said had targeted an Iranian force and "other Shiite militias" to prevent a planned drone attack on the Jewish state.
Nasrallah said those strikes killed two Hezbollah members but that the site hit was not a military facility.
"There were just Lebanese youth from Hezbollah in the place that was bombarded," he said.
Hezbollah, considered a terrorist organisation by Israel and the United States, is a major political actor in Lebanon and a key backer of the government in Syria's devastating civil war.