Former Test wicketkeeper Matthew Wade led the middle-order charge for as the visitors went to stumps on day two at 6-360.
Wade is unbeaten on 92, the only one of four Tasmanian batsman to have truly converted their starts, with Tom Rogers (28 not out) at the other end.
Wade was given a massive reprieve when he was dropped on 33 by Brendon Doggett, who spilled a regulation catch at point that proved costly for Queensland.
Coming to the crease with the visitors at 4-141, Wade piled on 80 runs with skipper George Bailey and another 84 with Simon Milenko to seize all the momentum.
Bailey was run out for 51 with the second ball after the tea break, foiled by a sensational direct hit by Michael Neser from mid-on.
That brought Milenko to the crease and the former Queensland all-rounder went straight to work, smashing five boundaries and two sixes to reach his 50 off just 54 balls.
But shortly after bringing it up, he was caught on the square leg boundary by Mitch Swepson to become Doggett's third scalp.
Doggett had figures of 3-81 off 23 overs with Jack Wildermuth and Michael Neser taking the other wickets.
Bailey won the toss and chose to bat first - a gutsy decision considering Tasmania need to secure an outright victory to win the title, after day one was abandoned without a single ball bowled because of a sodden outfield.
They faltered early, losing Alex Doolan for two, but a productive innings from Jordan Silk (76) put them in the ascendancy.
Silk hit 11 boundaries but was brought undone when he took a swipe at Doggett and top-edged it to fine leg, where Luke Feldman managed to dive underneath it and take a one-handed catch just centimetres off the turf.
Play will resume on Sunday at 9am.