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The overnight assault was one of the heaviest and most widespread for several months.
The army said a number of Hezbollah targets were hit, including what is described as the organisation's headquarters in Baalbek in the eastern Beka'a Valley. At least one raid took place close to the Lebanese capital, Beirut.
Israeli forces also shelled suspected Hezbollah targets in Southern Lebanon.
The Lebanese Information Minister, Anwar Khalil, has told the BBC of unconfirmed reports of civilian deaths in villages hit by Israeli forces.
Fearing revenge attacks, the Israeli army ordered civilians living along Israel's northern border to spend the night in air raid shelters. No attacks were reported, but the all-clear had not been given by dawn.
Prime Minister Binyamin Netanyahu was expected to convene members of his government on Monday morning to discuss the situation.
BBC Middle East Correspondent Paul Adams says Israel might be poised to launch a substantial attack along the lines of Operation Grapes of Wrath, a largely unsuccessful but extremely bloody Israeli operation three years ago.
On Sunday, Israel said it would wage "a ground, sea and air assault" in response to the Hezbollah attack, which killed an Israeli general, two other soldiers and an Israeli journalist.
However, Mr Netanyahu will be wary of following in the footsteps of his predecessor, Shimon Peres, who launched Grapes of Wrath on the eve of elections. The operation was widely criticised and he lost the poll.
Lebanese Prime Minister Selim Hoss has called for international pressure on Israel to stop the attacks, which he said were targeting civilians. The United States urged both sides to show restraint.
On Sunday evening, Mr Netanyahu said: "We have no desire whatsoever to escalate in Lebanon but neither can we sit aside and accept these criminal attacks."
Senior officer killed
The four Israelis died when a powerful remote-controlled bomb exploded shortly before midday (1000 GMT) on Sunday. They were in a convoy travelling near Israeli military headquarters at Marjayoun.
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One of the dead was Brigadier-General Erez Gerstein, the most senior Israeli officer to be killed in Lebanon for 17 years. He had been the chief officer in charge of liaison with Israel's local militia allies, the South Lebanon Army.
In Beirut, Hezbollah claimed responsibility for the attack and vowed to continue fighting Israel until it withdrew from Southern Lebanon.
Earlier in the week, three Israeli soldiers were killed and five wounded in a fierce battle involving helicopter gunships and warplanes.
Both incidents have sparked demonstrations in Israel by groups calling for a withdrawal from Southern Lebanon.
Israeli forces have occupied parts of Lebanon since 1978. Israel argues it needs a buffer zone to protect Israel from attacks by Islamic militants.
The Israeli Government recently raised the prospect of negotiating a withdrawal, but it wants the Syrian-backed Lebanese Government to agree to security guarantees first.
For their part, the Lebanese insist on an unconditional withdrawal in line with a 1978 UN resolution.
Country profiles
Israel threatens attacks inside Lebanon
(01 Jan 99 | Middle East)
'Victory or martyrdom': Hezbollah speaks out
(03 Dec 98 | Middle East)
History of Israel's role in Lebanon
(01 Apr 98 | Middle East)
Israel Defence Forces
Hezbollah
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