Larne dominated proceedings from the off in Serravalle and went close on 23 minutes when Dan Bent strode forward and had a low shot from the edge of the box saved.
New signing Montel Gibson then spurned a good opportunity to open the scoring on 39 minutes as he latched onto Lusty's low cross but saw his first-time effort turned around the post by goalkeeper Michele Nardi.
Aaron Donnelly and Chris Gallagher also flashed shots from outside the box wide before Larne eventually broke the deadlock on the stroke of half-time.
From a well-worked free-kick routine, Tomas Cosgrove's deep cross was headed back across goal by Sam McClelland and into the path of Lusty to sweep home from close range.
It was a slower start to the second half, with clear-cut chances at a premium until Larne thought they had doubled their lead on 77 minutes.
Substitute Josh Ukek won possession high up the pitch and fed Lusty, who coolly slotted beyond Nadi.
The goal was eventually chalked off after a VAR review with referee Artem Gasparyan deeming that Ukek had fouled Tre Fiori defender Alex Sirri in the build-up to the goal.