The Bulls were all out for 445 in their second innings, a lead of 269 runs, on the stroke of stumps on day three of the five-day Shield decider in Adelaide.
The highest run chase to win a Shield final is Victoria's 2-239 against NSW in 1990/91.
Clayton and Wildermuth struck contrasting but equally crucial centuries in a remarkable revival of Queensland's fortunes on Friday at Karen Rolton Oval.
Clayton's plucky 100 from 249 balls was followed by Wildermuth's free-flowing 111 from 140 deliveries.
"It's kind of yin and yang, the way I bat and he bats," Clayton said.
"He's obviously a lot more aggressive than me, he just comes out there and sees ball, hits ball.
"When he's on a heater, he's really on a heater and the way he batted today was fantastic. He played without fear."
Queensland had been on the ropes since being bowled out for 95 in their first innings, with a Jake Lehmann century steering SA to 271 all out in reply.
And the Bulls' chances appeared remote on Friday when at 6-221, just 45 runs ahead, Clayton and Wildermuth were united.
The pair turned the tide with a 103-run partnership, defying an SA attack led by ex-Queenslander Brendan Doggett, who has returned the best match bowling figures in a Shield final.
Doggett followed his 6-31 in the first innings with 5-109 for a combined 11-140. Tasmania's Shane Jurgensen (11-172 in 2001-02) is the only other bowler to take as many wickets in a Shield decider.
Clayton, who ran himself out for a golden duck in the first innings, reached his fifth first-class ton from 247 balls.
Two balls later, the left-hander was trapped lbw by spinner Ben Manenti (1-38).
Wildermuth, who went eight years between Shield centuries but has now scored two in two games, hammered SA's bowlers in a knock featuring 18 fours.
The 31-year-old found another ally in No.9 Mark Steketee, who made 51 and posted a match-high 122-run stand with Wildermuth, who was bowled by Nathan McAndrew (3-106) about 30 minutes before stumps.
Queensland are eyeing their first Shield title since 2020-21 while trying to extend SA's drought which dates back to 1995-96.
But the South Australians are hoping to collect the Shield and the one-day trophy in the same season for the first time, after their victory in the 50-over final against Victoria on March 1.
"We knew that Queensland weren't going to roll over," SA vice-captain Manenti said.
"We have believed all season that we can win from anywhere and this certainly isn't the furthest we've been behind the game, if we're behind it all.
"We have got 260-odd to chase and the Shield is ours. It won't be easy, but it's still a beautiful cricket wicket to bat on."
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