Trooper Shane Kearney, 23, was also seriously injured in the attack
Speaking on RTÉ's The Week in Politics, Minister for Enterprise Simon Coveney said the news was "more optimistic" about the condition of Trooper Kearney.
The soldier is now breathing independently and plans are being made to transfer him to Ireland when it is medically safe.
It is understood the soldiers' armoured vehicle was surrounded by a mob and fired on while travelling to Beirut at around 21:00 on Wednesday.
Three separate investigations are being conducted into the incident by the United Nations Interim Force in Lebanon, Lebanese authorities led by a military prosecutor, and the Irish Defence Forces.
Mr Coveney, who was the Irish defence minister until a cabinet reshuffle at the weekend, said they "will get to the bottom of what happened".
Mr Coveney said he had spent time with Pte Rooney's family who were "incredibly strong, impressive people".
Taoiseach (Irish prime minister) Leo Varadkar said Pte Rooney would be "offered full military honours which I think is appropriate".
"It is important we understand why an Irish soldier lost his life and another was severely injured," Mr Varadkar said.
"It is an immensely valuable mission. Thousands of Irish people have served there which is extraordinary when you think about it."