The strikes were launched around 4 am (0100 GMT), hitting both the Gaza Strip and southern Lebanon, an Israeli army statement said.
The military "struck targets including terror infrastructures belonging to the Hamas terrorist organization in southern Lebanon", it said.
It is the first time Israel has confirmed an attack on Lebanese territory since April 2022.
Explosions were heard by AFP journalists in Lebanon's Tyre region as well as the Gaza Strip, where Israeli air raids had begun before midnight.
AFP journalists near Tyre said they heard at least three explosions shortly after Friday's strikes, and a resident of a nearby Palestinian refugee camp said he "heard explosions".
"At least two shells fell near the camp", Abu Ahmad told AFP.
Tensions have soared between Israel and Palestinians during what is both the Jewish Passover and the Muslim holy month of Ramadan, prompting international calls for restraint.
The latest flare-up of violence comes after Israeli police clashed Wednesday with Palestinians inside Jerusalem's Al-Aqsa mosque -- Islam's third-holiest site.
On Thursday, the Israeli army said more than 30 rockets had been fired from Lebanese territory into Israel in the largest escalation on the northern border since Israel and Hezbollah fought a 34-day war in 2006.
This was followed by explosions in the Gaza Strip, heard by AFP journalists. The Israeli army confirmed it had hit two tunnels and "two weapon manufacturing sites" belonging to Hamas "as a response to the security violations of Hamas during the last few days."
Hours before Israel launched the air strikes into southern Lebanon, several missiles were fired from Gaza towards Israel.
The army said its air defences intercepted 25 of the rockets fired, while five hit Israeli territory.
After the pre-dawn strikes into Lebanon, the Israeli army said it "will not allow the Hamas terrorist organization to operate from within Lebanon and holds the state of Lebanon responsible for every directed fire emanating from its territory".
- 'A heavy price' -
Army spokesman Lt. Colonel Richard Hecht blamed Palestinian militants for the rocket fire from Lebanon.
"We know for sure it's Palestinian fire," he told reporters. "It could be Hamas, it could be Islamic Jihad."
Prime Minister Benjamin Netanyahu said: "Israel's response, tonight and in the future, will exact a heavy price."
Palestinian Islamist group Hamas, which rules the Gaza Strip, condemned the "appalling Israeli aggression" and said it "holds the Israeli occupation fully responsible for the consequences of such grave aggression".
The Palestinian health ministry said "partial damage" was done to the Al-Dorra children's hospital in Gaza City in the overnight strikes, condemning the "unacceptable" act.
Spokesperson Hecht said the Israeli army was aware of the Palestinian allegation and looking into it.
The United Nations Interim Force in Lebanon (UNIFIL), which patrols the area along the Israeli border, urged restraint.
"Both sides (Lebanon and Israel) have said they do not want a war," UNIFIL said in a statement, adding that force leader Major General Aroldo Lazaro was speaking with authorities from the two sides.
"The actions over the past day are dangerous and risk a serious escalation," UNIFIL warned.
Lebanon's caretaker prime minister, Najib Mikati, had on Thursday rejected any "escalation" from his country after the rocket attack.
Israeli emergency services reported one man lightly wounded by shrapnel and a woman injured while running to a shelter during the attack.
Inspecting his damaged office in the town of Shlomi, 46-year-old Shlomi Naaman told AFP: "I heard the siren, I heard the boom, I was in my home, it was very, very scary."
- Mosque raid -
Israeli riot police had on Wednesday stormed the prayer hall of Al-Aqsa mosque in a pre-dawn raid aiming to dislodge "law-breaking youths and masked agitators" they said had barricaded themselves inside.
The violence sparked an exchange of rockets and air strikes with militants in the Gaza Strip.
The US said it recognised "Israel's legitimate right to defend itself against all forms of aggression", State Department spokesman Vedant Patel told reporters.
UN chief Antonio Guterres called on "all actors to exercise maximum restraint".
Lebanese militant group Hezbollah had warned it would support "all measures" that Palestinian groups may take against Israel after the clashes.
The Lebanese group has close ties with Hamas and with Islamic Jihad, another Palestinian militant group based in the Gaza Strip.
The rocket fire came a day after Hamas leader Ismail Haniyeh arrived in Lebanon for a visit.
Haniyeh said late Thursday that the Palestinians would not "sit with their arms crossed" in the face of Israeli "aggression" against Al-Aqsa.
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