Western Australia 299 and 3 for 55 need 66 more runs to beat South Australia 128 and 291 (Hunt 108, Carder 68, Paris 5-63)
Paris snared 5 for 63 to dismiss SA for 291 in their second innings at the WACA Ground on Friday, leaving WA with a victory target of just 121.
But there could still be a final twist to the match, with WA reduced to 3 for 55 by stumps on day three in a nervous start to the run chase.
Openers Cameron Bancroft and Sam Whiteman both fell cheaply, but the biggest setback came when star veteran Shaun Marsh was run out for 16 in disastrous fashion to leave WA in trouble at 3 for 39.
Cameron Green set off for a quick single after hitting the ball to point, but a direct hit from Hunt with only one stump to aim at caught Marsh several metres short of the crease. Green and Hilton Cartwright made it safely to stumps, with WA requiring a further 66 runs to win on a tricky deck.
Hunt top-scored for SA with 108 off 245 balls, while Jake Carder (68) and opener Jake Weatherald (44) also posted handy totals.
Redbacks had been in huge trouble after WA posted 299 in reply to SA's meagre first-innings total of 128. But they hit back during the first half of the day on Friday, with Hunt's heroics helping the visitors reach 1 for 212 at one stage.
However, just as SA looked set to build a respectable lead, the wheels fell off completely as they lost 9 for 79. Paris was the chief destroyer, with the 28-year-old snaring the scalps of Hunt, Weatherald, and Jake Lehmann among his haul.
Cartwright took the key wickets of Travis Head, superbly caught in the deep by D'Arcy Short, and Carder while Jhye Richardson toiled hard for a return of 2 for 75 from his 21.3 overs.
Paris has taken four metres off his run-up in a bid to avoid the injuries that have plagued his career, and the tweak is already paying dividends
"These are the moments you enjoy and it's the reason why you do all the time on the sidelines, in the gym, or rehab - because these moments outweigh all the tough days," said Paris, who played two ODIs for Australia in 2016 before suffering numerous injury setbacks.
"I love days like this, love being able to contribute to WA, and hopefully there's a few more to come."