Lebanese security officials said two civilians were wounded when Israel responded with artillery fire and air strikes.
One of them - reported to be an 82-year-old man - later died during surgery to remove shrapnel from the Israeli shelling, according to relatives and medical sources.
The clash, in the disputed Shebaa Farms area, is the first such incident in several months.
The Israeli army said it had suffered no casualties in the shelling.
Hezbollah has clashed sporadically with Israeli troops since Israel pulled out of southern Lebanon in 2000.
Hezbollah says the Israeli withdrawal from Lebanese territory was incomplete, claiming the Shebaa Farms area for Lebanon - a claim Israel rejects.
Lebanese police said guerrillas fired anti-tank rockets and artillery rounds at Israeli positions at Roweissit al-Aalam and Tallet Sammaqa, opposite Kfar Shuba in southern Lebanon.
Hezbollah itself said that it had attacked Roweissit al-Aalam after spotting Israeli soldiers leaving their position.
According to the police, Israeli fighter-bombers fired four missiles at an area to the west of Kfarshuba village.
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