It was the first clash between the two sides in five months.
Israeli military sources said it was the Shi'ite militant group which sparked the fighting with an attack on Israeli troops.
But Hezbollah said its fighters attacked the Israeli patrol only after it strayed across the UN-demarcated frontier into Lebanese territory.
"The Islamic Resistance confronted an Israeli force composed of two military vehicles that had penetrated the Lebanese side of the village of Ghajar," a Hezbollah statement said.
"A violent exchange ensued leading to the destruction of the two vehicles, and dead and wounded in enemy ranks, as well as three dead among the resistance fighters," it continued.
Ghajar lies partly in Lebanese territory and partly in the occupied Golan Heights, a strategic plateau which Israel seized from Syria in 1967.
Israeli television said the Hezbollah barrage was a cover for a kidnap attempt on Israeli soldiers from Ghajar, but that the attempt was thwarted.
Fighting escalates
The deadly clashes prompted Hezbollah to launch a massive bombardment of Israeli positions in the disputed Shebaa Farms border district.
Lebanese police said Hezbollah fired 300 shells in an hour into the water rich area, where Israeli sources said some soldiers were wounded.
The Shebaa Farms have remained a source of tension ever since Israel ended its 22-year occupation of south Lebanon in 2000.
The area was captured by Israel from Syria in 1967 but is now claimed by Lebanon with Syrian consent, an arrangement that is not recognised by the United Nations.
Hezbollah also fired Katyusha rockets over the border into Israel.
Israeli civilians were hit by rocket fire in the border towns of Kiryat Shmona and Metulla, where residents were advised to take cover in bomb shelters.
Strong retaliation
Israel retaliated with air strikes against suspected Hezbollah targets in south Lebanon and an artillery bombardment of the border area.
"Following attack by Hezbollah, the air force attacked a command post and a number of access routes in southern Lebanon," an Israeli army spokesman said.
Israeli aircraft fired missiles on three suspected Hezbollah positions, one on the outskirts of Ghajar and two some distance away - south of the village of Khiam, and southeast of the port city of Tyre, Lebanese police said.
In the evening, an Israeli helicopter launched a missile attack on a suspected Hezbollah position near the coastal town of Naqura, where the UN peacekeeping force has its base.
Israeli aircraft also flew over a large swathe of southern Lebanon, prompting Hezbollah anti-aircraft batteries to open up, police said.
By evening the clashes had largely subsided but Lebanese police said they were the fiercest in months – raging for three hours.
Tensions rise
Tension has risen on the border since the Israeli army launched a massive artillery bombardment from the Shebaa Farms two weeks ago.
There has been a sharp intensification of Israeli overflights since, in defiance of UN calls for an end to violations of Lebanese air space.
Last year, UN Security Council resolution called for the disarmament of Hezbollah.
The organisation has refused, and has so far been supported by the Lebanese government - which claims Hezbollah is not a militia but a movement resisting Israeli occupation.
Lebanon has also rejected repeated UN and US demands to deploy the army along the country's southern border, saying it will not serve as an Israeli protection force.