The Reds led 24-3 midway through the second half, but in a replay of last season's Suncorp Stadium clash, had to withstand a late Drua flurry and somehow hung on.
Josh Flook should have had a first-half hat-trick, dropping a brilliant short kick from James O'Connor as the line beckoned after already crossing twice.
And Taj Annan was slick in his starting debut at No.12, setting up two tries in the second half and defending well in place of concussed Wallabies centre Hunter Paisami.
Those slick moments were few and far between in a sloppy Sunday afternoon performance that featured 32 penalties blown by referee Reuben Keane.
Still, Queensland seemed to be cruising when Annan's cut-out pass put Suliasi Vulivanu over for a 21-point lead.
But the Drua, relishing the steamy conditions, found some space and Leone Rotuisolia and Kalaveti Ravouvou raced over.
They looked destined to roll through the Reds again for a match-winner, before try-scorer Harry Wilson dragged down a defender and winger Filipo Daugunu became the unlikely man to steal possession with a game-winning defensive play.
The result leaves both teams at 2-2, with the Reds' next task in Melbourne against the Rebels.
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