A four-game losing streak had threatened to derail the Roar's league campaign, but they were in fine form on Thursday at Suncorp Stadium, where Lustica was the star of the show.
The 23-year-old scored twice before the break, with Andrija Kaluderovic having broken the deadlock.
Devante Clut's simple finish continued the rout in the second half, while Brandon Borrello added the fifth before the visitors finally got off the mark through Daniel Bowles' own goal.
Lustica completed his treble with 11 minutes to go as the hosts moved to within five points of sixth-placed Melbourne City.
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