The attack sparked fierce exchanges of rifle fire and grenades between Lebanese soldiers and Jund al-Sham members.
They said two people were wounded and scores of people, with their belongings in plastic bags, were fleeing the scene.
The army later brought in reinforcements and deployed armoured vehicles mounted with heavy machine guns.
Killed by sniper
Earlier, the ANI official news agency in Lebanon reported that at least three other Fatah al-Islam fighters were killed in Nahr al-Bared camp on Sunday.
Security sources said that two army soldiers had been killed overnight, bringing the number of troops killed since Friday to eight.
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Zeina Khodr, Al Jazeera's correspondent, reporting from outside Nahr al-Bared camp, said Fatah al-Islam confirmed that Abu Riyadh, one of its leaders, was killed by an army sniper on Saturday.
She said there had been conflicting reports in previous days that other senior Fatah al-Islam leaders had been killed, reports that were denied by the group.
Machinegun fire could be heard outside the camp on Sunday as the army continued its siege backed by helicopter gunships and tanks.
The precise death toll inside the camp is not known.
Explosions rocked the camp but witnesses said the fighting was less intense than the previous two days.
As the siege continued, Abu Salim Taha, a spokesman for Fatah al-Islam, claimed that troops from Unifil, a multi-national peacekeeping force whose remit is largely concerned with monitoring southern Lebanon, had taken part in the shelling of Nahr al-Bared camp.
A deputy spokesperson for Unifil denied the allegations.
Yapmina Bouzaine said: "These claims are utterly unfounded. Unifil's maritime task force have no part whatsoever in the developments in and around Nahr al-Bared camp."
She said the maritime task force was acting within its original mandate, assisting the Lebanese authorities in preventing the illegal flow of arms via the sea.
Al Jazeera's Zeina Khodr said the Lebanese army had managed to reach a few hundred metres inside the camp and clear out pockets of fighters.
More than 16 people, fighters and civilians, have died in the camp over the weekend. Fatah al-Islam said it has lost three fighters.
The Lebanese government has said it aims to eliminate Fatah al-Islam once and for all.
Fouad Siniora, the Lebanese prime minister, said on Saturday that the only option for members of the group was to give up their weapons and surrender.
"This is a terrorist gang," he said in a TV interview.
"They have to surrender themselves and their arms."
But Abu Salim Taha said: "There is no way we will give up our weapons because it is our pride. We cannot even contemplate surrendering."
Military helicopters have fired rockets and machinegun barrages at targets on the camp's coastal side.
"Since yesterday morning, the shelling has been ongoing all over the camp. Two shells fell on the building I'm in now. Several buildings have collapsed," a Palestinian resident inside the camp said.
"There's only one clinic with one doctor left. There's no electricity, bread or medicine."
Elderly and disabled
Al Jazeera's Zeina Khodr quoted humanitarian organisations as saying on Sunday that the majority of refugees remaining in the camp were elderly or disabled.
|Lebanese soldiers stake out the Nahr al-Bared|
camp as part of a continuing siege [AFP]
She said: "Approximately 150 people are in wheelchairs ... Since the [latest army] offensive began on Friday, no relief supplies have made their way in to the camp."
The army has urged refugees still trapped inside the camp to "be patient and to expel those criminals from among you".
Siniora said on Saturday that the camp's population had fallen from more than 31,000 to fewer than 3,000, including the fighters, after civilians fled after the outbreak of fighting on May 20.
He accused Fatah al-Islam of preventing the remaining civilians from leaving the camp in an apparent bid to use them as human shields.