Queensland 175 and 2 for 65 (Clayton 34*, Street 16*, McClure 1-7) need 285 runs to beat Victoria 307 & 217 (Perry 75, Short 58, Bartlett 4-32)
Street and Clayton then thwarted Victoria's late charge with a gritty half-century partnership to revive Queensland's hopes.
Victoria, however, remain in the box seat in what has been a low-scoring contest, but their bid for victory could be impacted by possible inclement weather in Melbourne on Sunday.
As the Shield season resumes after a two-month break, with four of ten rounds left, Victoria can leapfrog second-placed Queensland with an outright win. They gained a stranglehold with early strikes starting with Burns being cleaned bowled by quick Fergus O'Neill for 6.
It capped a disappointing comeback for Burns, who made only two runs in Queensland's first innings after returning from a hamstring injury sustained in the BBL season opener.
Clayton made the most of his good fortune to combine with Street, who faced 102 balls for his 16 runs. Scoring was difficult against a disciplined Victoria attack, but they eventually reduced Queensland's target on the final day to 285 runs.
Queensland slowly clawed back into the contest after being on the back foot for most of the match, having trailed in the first innings by 132 runs.
Victoria started day three on 2 for 21 in their second innings and lost the early wickets of opener Travis Dean and Campbell Kellaway.
But nightwatcher Perry made a career-best 75 and combined in a century partnership with Matthew Short, who struck a measured half-century in a contrast to his belligerent batting during the BBL.
Left-handed Perry has a liking for the nightwatcher role, having played several indefatigable knocks previously, including 74 against WA last season.
He eyed a maiden century before slicing to backward point off quick Mark Steketee, who was finally rewarded for his toil.
Victoria lost a slew of wickets in an attempt for quick runs and were bowled out for 217 on the stroke of tea. Emerging allrounder Xavier Bartlett claimed the spoils for the Queensland attack with 4 for 32 from 14 overs.
With five players backing up from Brisbane Heat's barnstorming BBL campaign, which ended in a painful final's defeat to Perth Scorchers a week ago, Queensland have looked lethargic at times but will need to summon a mighty batting effort on the final day to avoid defeat.
Tristan Lavalette is a journalist based in Perth