Authorities accuse Awad of having "incited" rebels to carry out attacks two years ago in the port city of Tripoli, near the Palestinian camp, that killed 21 people, including 13 soldiers.
Those found guilty of incitement to carry out deadly attacks can face the death penalty under Lebanese law.
Awad, one of the most wanted fighters in Lebanon, opened fire at troops along with his comrade and the soldiers responded killing the pair, the spokesman said.
The clash broke out in the Bekaa Valley town of Chtaura and both men were travelling on false identities, the army said.
Earlier, the spokesman said the army had been pursuing the pair since they emerged from another Palestinian refugee camp, Ain al-Hilweh in south Lebanon, but he did not give a timing.
Several rebel groups are suspected of having taken refuge in the north and east of the country, and in the 12 Palestinian refugee camps scattered across Lebanon of which Ain el-Hilweh is the largest.
Three year hunt
Awad had been sheltering in Ain el-Hilweh for more than a year, according to the army. Authorities also charge that the wanted fighter was monitoring the movements of Lebanese army troops as well as of UN peacekeepers stationed in south Lebanon.
Lebanese authorities suspected Awad of being the main leader of the Fatah al-Islam group.
Members of the group arrested in Syria said in testimony aired on Syrian state television two years ago that Awad took over the mantle from the group's former leader Shaker Abssi, whose whereabouts remain unknown.
Abssi - who appears to have fled Lebanon - and a third member of the group, Lebanese citizen Abdel Ghani Jawhar, also figure among the top wanted fighters.
Fatah al-Islam has been linked to deadly attacks that targeted senior army and police officers in December 2007 and January 2008 respectively, as well as three UN peacekeepers in south Lebanon in June 2007.
In August 2007, the US state department designated Fatah al-Islam, which denies formal links with al-Qaeda, as a "terrorist" group.