The hosts beat Poland 3-1 in Brisbane on Sunday, de Minaur sealing the tie with an impressive 6-3 6-3 defeat of top-50 talent Kamil Majchrzak after Purcell and Rinky Hijikata combined to clinch a pivotal point in doubles.
Their wins came after de Minaur and James Duckworth had won and lost their singles matches on Saturday respectively, sending Australia into the Cup's penultimate round early next year.
Victory there would qualify them for next November's finals in Italy, with Australia chasing a 29th Davis Cup title but first since 2003.
With his serve firing and backhand on song, it was a therapeutic response to de Minaur's disappointing second-round US Open exit earlier this month.
His job was made easier by Purcell's brilliant Davis Cup showing as the 28-year-old and former world No.40 returns from a controversial 18-month doping ban.
"Max's first tie on home soil; he's grown up dreaming of getting the opportunity and he got screwed over," Australian captain Lleyton Hewitt said in reference to the controversial suspension.
The two-time major doubles champion initially copped a two-year ban after he was found to have a number of vitamins - all of which were on the approved list - in his system above the legally permitted levels.
Purcell said it was the result of an IV infusion after he fell sick while holidaying in Bali, and the ban was reduced because of his co-operation.
A low-profile return in lower-level tournaments was blown open on Sunday with a brilliant tiebreak sequence that sent Australia hurtling towards victory.
Purcell drilled a backhand return to win the opening point on the Polish serve, then produced a pair of aces to shoot the hosts clear 3-0.
An unforced error from Poland was followed by two more Purcell winners, and he then held his nerve on a series of low volleys to eventually clinch the set.
Hijikata then held impressively and, after missing two break-point chances, nailed a forehand return to nab the match's only service break.
Purcell talked down his impact, leaving partner Hijikata to do it justice.
"Joke," he grinned. "Unreal level from him."
"And from then on he didn't miss a return, and I don't know how many times they touched his serve.
"I was just standing out there; he made my life very, very easy and it was sick watching 'Demon' do his thing. A nice day to be a part of."
With his backhand and serve firing, de Minaur followed up his 57-minute annihilation of Maks Kasnikowski on Saturday with another fine performance.
The win was an impressive one considering Majchrzak, inside the world's top 50 earlier this year, beat de Minaur in the 's-Hertogenbosch final when they last met in June.
Australia's win followed a shock loss on the road to Ecuador earlier this year that plunged them into the must-win encounter against Poland.
They still need to win next year's qualifying round, against an opponent still to be decided, to progress to the finals in Italy in November, 2027.
Polish captain Mariusz Fyrstenberg tipped his cap to de Minaur and Purcell in particular.
"Purcell looked like he didn't have a break. Very good," he said.
"They just stepped up even higher (in the tiebreak), played four great doubles points and it was tough from there."
Fyrstenberg called out former world No.6 Hubert Hurkacz, who played the US Open and is scheduled to play in Chengdu from Tuesday but skipped the Davis Cup tie as he manages a back injury.
"We're good enough to go up, definitely we miss Hurkacz, who doesn't want to play and I hope it's going to change," Fyrstenberg said.
"It's quite frustrating; others sacrifice tournaments, fight for everything, when our top guy is not playing and he's playing singles and doubles next week."
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