"The army has found the bodies of six Islamists inside an apartment building in Abu Samra," a spokesman for the Lebanese army said.
The clashes in the Abu Samra neighbourhood of Tripoli began after Lebanese army troops raiding a building came under fire.
"The Palestinian refugees are not being treated properly by Lebanon"
Send us your views
"[Abu Samra] is an area where most Islamist groups have their bases," Zeina Khodr, Al Jazeera's correspondent in Lebanon, said.
Three Saudi nationals, one ethnic Chechen and two Lebanese were among the fighters killed by Lebanese army forces, officials said.
The dead fighters were not members of Fatah al-Islam, whose fighters have battled the Lebanese army in recent weeks at Nahr al-Bared camp, security sources said.
It is understood that information from a captured Fatah al-Islam member led the army to the apartment where the shootout took place.
A spokesman from Australia's Department of Foreign Affairs and Trade said three men with Australian citizenship were arrested by Lebanese security forces in Tripoli, without giving further details.
However, Lebanese security officials confirmed only one man with Australian citizenship was arrested in Tripoli in recent days.
Battles at camp
The Lebanese army has been fighting Fatah al-Islam for more than a month at the nearby Nahr al-Bared refugee camp.
At least 176 people have been killed in the fighting.
It is unclear how many civilians have died inside the camp as security forces are barred from entering Lebanon's 12 Palestinian refugee camps by a decades-old Arab agreement.
"There are real fears in Lebanon that the crisis may spill out of the refugee camp," Khodr said.
"We do know that many officials have warned that there could be a rise in the presence of al-Qaeda in Lebanon."
Palestinian mediators held talks on Friday with Fatah al-Islam at the camp and were due to hold more talks over the weekend to find a solution acceptable to the army and Fatah al-Islam.