Saturday's deaths raised to 96 the number of soldiers killed since fighting between the army and Fatah Islam fighters began on May 20 in the Nahr el-Bared camp on the outskirts of the northern port city of Tripoli.
The National News Agency said on Saturday that the army was making "important progress" and "tightening the noose on the Fatah Islam gang until it is eliminated".
The army said it had seized control of a number of buildings that had been used by Fatah Islam fighters to harbour snipers and was clearing the buildings of mines on Friday.
"The Palestinian refugees are not being treated properly by Lebanon"
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In an apparent attempt to expand the conflict zone outside the camp, Fatah Islam fighters launched 19 Katyusha rockets on Friday that hit villages next to the camp, injuring two people.
Lebanese officials claimed victory on June 21 after soldiers seized Fatah Islam positions on the camp's peripheries, but the fighters retreated into an area of densely packed buildings whilst continuing daily fire fights.
The army lost six soldiers on Thursday in one of the heaviest bombardments against the fighters. But the army has denied it is conducting a final assault against Fatah Islam, who have vowed to fight to the death rather than surrender.
At least 60 fighters and more than 20 civilians have been reported killed in the fighting, the country's worst internal violence since the 1975-90 civil war. The camp housed more than 30,000 Palestinian refugees before the battles began.
Most of the camp's residents have already fled, but a few thousand are thought to have stayed in their homes.