Russia, spearheaded by world number five Daniil Medvedev won both their singles to sweep past Argentina.
Kyrgios and De Minaur stepped in late to replace doubles specialists John Peers and Chris Guccion in what was a controversial call from captain Leyton Hewitt, and the gamble appeared to have failed as the pair lost the first set 6-3.
Kyrgios gave Australia the lead with a 6-2, 6-2 win over Cam Norrie in the opening singles, but Evans upset No. 18-ranked de Minaur 7-6 (4), 4-6, 7-6 (2) to level the match.
Kyrgios held serve. 10-9 Australia.
Admitting to being flat during the deciding rubber, Australian No.1 de Minaur said Kyrgios "carried" him during the match before literally doing so after a tense win where they saved four match points.
"That was incredible tennis".
With the crucial quarter-final pitching the "pommies" against the hosts, the best of three clash kicked off with two singles matches.
"Really proud of the boys, the way they were able to turn it around.it could have gone either way".
The young Australian, who won three ATP titles past year, kept himself in the match by holding on to an early break to take the second set. It paid off after a seesawing 3-6, 6-3, 18-16 victory, sealed in a gripping match tiebreaker.
With Russian team captain Marat Safin sitting in his corner, Medvedev broke his opponent's serve twice - and gave one break back - in an entertaining first set, where his slapping forehand ebbed between sublime and out-of-control.
A time violation for Evans on his serve - given when it looked like he had already started his service motion - may have been playing on his mind to allow De Minaur a break back courtesy of a passing shot that was helped by the net cord.
Hewitt said the decision was in the best interests of the team, explaining that he could not leave Kyrgios on the bench because of the way he had been serving, and knew de Minaur would respond positively in the doubles. "All the matches were won in singles, which is wonderful, and I think only Spain did the same so far", Medvedev said.
Rafael Nadal vs. Nick Kyrgios to decide it all?
It is partly due to having a six-month suspended ban hanging over him, imposed in September for calling an Irish umpire a 'potato and a 'tool in Cincinnati.
He also accused ATP become corrupt before backtracking on that allegation.
"When he had a short backhand I thought that was it, I can't believe he missed that, I'm sorry", Kyrgios said.
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