"What happened is a declaration of war," Michel Aoun told Jan Kubis, the UN Special Coordinator for Lebanon, in a meeting.
"This allows us to resort to our right to defend our sovereignty, independence and territorial integrity," he added, in a statement released by his office.
The army said two Israeli drones had violated Lebanese airspace over Beirut before dawn on Sunday, with one exploding in the air.
Hezbollah movement chief Hasan Nasrallah said an armed drone had hit a target in the party's Beirut stronghold, without specifying.
It was the first such 'hostile action' since a 2006 war between Hezbollah and Israel, added the head of the Iran-backed party, also vowing to retaliate.
"I have repeatedly said before that Lebanon will not fire a single shot from its border unless it is in self-defence," said Aoun, an Hezbollah ally.
"What happened yesterday allows us to exercise this right," he added.
Earlier on Monday, Israeli air strikes targeted a Palestinian faction loyal to the Syrian government in eastern Lebanon.
Hezbollah chief threatens Israel after Beirut 'drone attack'
State-run NNA news agency said "three hostile strikes" after midnight hit Lebanon's eastern mountains near Qusaya town "where the Popular Front for the Liberation of Palestine-General Command has military posts".
Aoun said Israeli attacks on Qusaya and Beirut violated a UN Security Council resolution ending the month-long 2006 war that killed 1,200 people in Lebanon, mostly civilians, and 160 in Israel.
Lebanon had filed a complaint against Israel to the UN Security Council.
On Sunday, Nasrallah threatened Israel after the Beirut attack.
"The time when Israeli aircraft come and bombard parts of Lebanon is over," he said in a televised speech.
"I say to the Israeli army along the border, from tonight be ready and wait for us," he added. "What happened yesterday will not pass."
Hezbollah, considered a terrorist organisation by Israel and the United States, is a major political actor in Lebanon and also a key government backer in war-torn Syria.
The latest incident also came after Israel on Saturday launched strikes in neighbouring Syria to prevent what it said was an Iranian attack on the Jewish state.
Nasrallah on Sunday said two Hezbollah members were among those killed in the strike. They were laid to rest in Beirut's southern suburbs on Monday, amid a large turnout of party supporters.
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