De Minaur and Kyrgios saved four match points before beating Jamie Murray and Joe Salisbury 3-6 6-3 18-16 in Sydney to seal a 2-1 victory for the hosts.
That decisive win came after Dan Evans kept Great Britain alive with a 7-6 (7-4) 4-6 7-6 (7-2) defeat of De Minaur at Ken Rosewall Arena.
Kyrgios, who beat Cameron Norrie to put Australia in front, lifted his team-mate on his shoulders and paraded around the court after they set up a showdown with Spain or Belgium on Saturday.
"t was unreal. The adrenaline has kind of worn off and I'm exhausted after that match. It was awesome," Kyrgios said.
"Today was probably one of the best moments in my career, definitely. Just making it through to the semi-final, the first-ever ATP Cup in that type of fashion, was pretty special.
"We genuinely care for each other, and [our team-mates] were genuinely just so ecstatic to get through."
Russia remain unbeaten after Daniil Medvedev and Karen Khachanov secured a 2-0 triumph over Argentina.
Khachanov put Russia ahead with a 6-2 7-6 (7-4) win over Guido Pella and Medvedev delivered the victory by seeing off Diego Schwartzman 6-4 4-6 6-3.
Medvedev was given a point penalty for clattering the chair umpire's chair with his racket but regained his composure following a chat with captain Marat Safin.
Serbia or Canada stand in the way of Russia and a place in the final.
Garin, the third seed at the ATP 250 event played on clay, rushed past Hungarian Attila Balazs 6-3 6-0 in just 68 minutes in Argentina.
The Chilean incredibly lost just eight points on serve, while converting five of 16 break points in a dominant performance.
Garin will face Uruguayan sixth seed Pablo Cuevas, who survived a test to get past Italian Gianluca Mager 6-7 (5-7) 7-6 (7-2) 6-1.
Pella, the second seed and last year's runner-up, fell to Frenchman Corentin Moutet after a gruelling contest.
Moutet beat Pella at the French Open last year and he repeated the feat courtesy of a 6-7 (4-7) 7-5 6-3 victory after just over three hours on court.
Slovakian Andrej Martin will face Moutet after overcoming Spanish qualifier Carlos Taberner 6-3 7-6 (9-7).
Nishikori trailed 6-4 4-2 and was forced to break serve as Guido Pella attempted to close out the match in the second set, before sensationally rallying to a 4-6 7-6 (7-4) 6-2 win on Monday.
A two-time champion at the ATP 500 tournament on the Barcelona clay, Nishikori broke serve on three occasions in the final set en route to the second round.
"He had the match today at 6-5, serving [for the match]," said Nishikori. "He was much better in the first and second [sets] and 3-0 up in the tie-break. I don't know how I fought through. [In the] third set, I was playing much better.
"I was making so many unforced errors in the first and second [sets]. Maybe he got a little bit tired and I was playing strong in the third set. I want to take that to the next round… The way I fought today and the way I played in the third set was good."
Next up for 2014 US Open finalist Nishikori is 13th seed Cristian Garin.
Ilya Ivashka topped Tallon Griekspoor 6-3 6-1 and the qualifier's reward is a showdown with top seed and 20-time grand slam champion Rafael Nadal.
Cameron Norrie, Pablo Andujar, Jeremy Chardy, Pierre-Hugues Herbert, Jaume Munar, Federico Gaio, Egor Gerasimov and Bernabe Zapata Miralles all advanced to the last 32.
At the Serbia Open, seeds Filip Krajinovic and John Millman booked their spots in the round of 16.
With the ATP 250 tournament being staged for the first time since 2012 and headlined by world number one Novak Djokovic, Serbian fifth seed Krajinovic overcame Stefano Travaglia 7-5 5-7 6-3 in a Belgrade rollercoaster.
Millman – the Australian seventh seed – took down local wildcard Danilo Petrovic 7-5 6-4 as Marco Cecchinato and Nikola Milojevic won through.
Nishioka was the highest seed in action on the first day of the tournament, and he had his back up against the wall after failing to win a game in a troubling second set, but produced three breaks of serve in the back-and-forth decider.
Meanwhile, Ecuador's Emilio Gomez earned a shot at top seed and world number seven Taylor Fritz after advancing 7-6 (7-3) 6-4 against Taiwan's Tung-lin Wu.
Portugal's Nuno Borges continued his winning run after earning his spot through the qualifiers, eliminating America's Steve Johnson 6-4 3-6 6-4, but the crowd got something to cheer for when the USA's Denis Kudla got the better of Australia's Jordan Thompson 3-6 6-4 7-6 (7-2).
Further south at Argentina's Buenos Aires Open, Serbia's Laslo Djere booked a blockbuster showdown against world number two Carlos Alcaraz after repelling the challenge of Italy's Fabio Fognini 6-4 6-4.
Argentinian Pedro Cachin put on a show for his hometown fans with a 7-6 (7-2) 6-4 victory over Brazil's Thiago Monteiro, but Cachin's compatriot Guido Pella had less success with a 6-4 7-5 defeat at the hand of Spain's Jaume Munar.
Thiem, last in action in an Australian Open final loss to Novak Djokovic, returned with a 6-2 4-6 6-1 victory over wildcard Felipe Meligeni Alves on Tuesday.
The 2017 champion of the ATP 500 event played on clay in Brazil, top seed Thiem grabbed two breaks of serve in the deciding set.
Having started 2020 with four losses in five matches, Borna Coric found a victory by overcoming Juan Ignacio Londero 7-6 (7-5) 7-5.
Dusan Lajovic joined Thiem and Coric as the three seeds to win, but two fell in the opening round.
Casper Ruud, coming off a title win at the Argentina Open, went down to qualifier Gianluca Mager 7-6 (7-4) 7-5.
Guido Pella, the 2016 runner-up, was edged by Thiago Monteiro 5-7 6-4 7-6 (7-3) after just over three hours.
At the Delray Beach Open, Sock claimed his first non-Laver Cup singles win since late 2018.
The American 2017 champion, a wildcard, battled past last year's winner Radu Albot 3-6 6-3 7-6 (7-2), saving a match point in the final set and breaking down in tears following his victory.
Canadian second seed Milos Raonic brushed past lucky loser Denis Istomin 6-2 6-2 and Reilly Opelka served 20 aces in a 6-7 (5-7) 6-4 7-6 (7-4) victory over Ernests Gulbis.
Ugo Humbert, the French sixth seed, battled through, while third seed Taylor Fritz fell to qualifier Cameron Norrie 6-4 6-7 (4-7) 6-4.
Thompson overcame big-serving American Isner – who had a first-round bye – 7-6 (7-2) 6-7 (3-7) 6-3 in New York on Thursday.
Isner was playing his first match since being forced to retire from his third-round clash against Stan Wawrinka at the Australian Open due to a foot injury.
Thompson lost to Isner in last year's New York quarter-finals but the Australian turned the tables after two hours, 31 minutes.
"I'm pretty stoked about that. That was a tough match," Thompson said. "I didn't get too many looks at Isner's serve, but managed to get the one break and held onto it."
Standing in the way of Thompson and the semi-finals is Andreas Seppi.
Opelka opened his title defence by cruising past Yoshihito Nishioka 6-4 6-4 at the ATP 250 tournament.
Next up is qualifier Jason Jung, who stunned seventh seed Cameron Norrie 6-4 6-4, Kyle Edmund – the eighth seed – was too good for German Dominik Koepfer 6-2 6-4.
At the Argentina Open in Buenos Aires, Diego Schwartzman and Guido Pella moved through to the quarters on clay.
Schwartzman – the top seed – prevailed 6-3 4-6 6-2 over Federico Delbonis and second seed Guido Pella survived a marathon against Facundo Bagnis 7-6 (7-2) 6-7 (2-7) 6-4.
Fellow seeds Laslo Djere and Albert Ramos-Vinolas, however, crashed out following defeats to Juan Ignacio Londero and Pablo Cuevas.
Fognini, competing for the first time since undergoing arthroscopic surgery on both ankles in May, went down 6-1 6-2 to world 303 Huesler in a little under an hour.
The 2011 French Open quarter-finalist was broken in the second game and Huesler, who had just two previous career wins on the ATP Tour, broke again in the sixth.
After saving three break points in the early stages of the third set, Huesler held serve and secured a famous win in Kitzbuhel with minimal fuss.
Feliciano Lopez awaits in the last eight after battling to a 7-5 6-7 (4-7) 7-6 (7-3) victory over seventh seed Guido Pella in three hours.
