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.
Organisers confirmed on Sunday that the WTA event and the ATP Indian Wells Masters, which were both scheduled to start on Monday, had been called off due to concerns over the coronavirus.
The Riverside County Public Health Department declared a public health emergency on Sunday after a confirmed case in Coachella Valley.
World number 77 Flipkens reacted with angry surprise to a tweet from the official tournament account that confirmed it had been called off, saying: "And the players had to find out... through Twitter...".
When American player Nicole Gibbs replied to tell Flipkens "There was an email", the Belgian responded: "I saw it on Twitter first."
She also sent a message to the WTA and urged fellow professionals to share it, saying: "@WTA isn't the least you can do is [sic] organising an emergency meeting with the players????"
Rising American star Coco Gauff said she was "so sad" that the event had been cancelled. The 15-year-old had been due to make her debut at the event this week.
"Safety is always the no.1 priority," she tweeted.
Meanwhile, Jamie Murray expressed concern over the rest of the 2020 calendar given the Indian Wells tournaments were called off due to just one confirmed coronavirus case.
"Doesn't bode well for the tour if IW cancelled for 1 confirmed case in Coachella Valley," he wrote on Twitter. "Broward county (Miami Open home) has more confirmed cases. Monte Carlo borders northern Italy currently in lockdown. Rome Masters? French Open? Wimbledon?!!!"
In a statement following the Indian Wells cancellations, WTA chairman and CEO Steve Simon said: "It is too soon to speculate about what will happen to other tournaments that follow.
"We will continue to closely monitor the situation. Health and safety will always come first."
Seven-times grand slam doubles champion Murray was asked about the decision to grant Djokovic a medical exemption for the year's first grand slam tournament at Melbourne Park.
Players have been obliged to either prove they are vaccinated against COVID-19 or apply for an exemption, which tournament organisers said involved a "rigorous" process to prove eligibility.
World number one Djokovic is a nine-time Australian Open men's singles champion, yet it was unclear until Tuesday whether he would be allowed to play the event this year.
There has been early backlash to the decision to clear him, with Melbourne having been hit by multiple lockdowns during the pandemic and many believing only vaccinated players should be playing.
Murray, brother of Djokovic's long-time friend and rival Andy Murray, spoke after the Serbian was permitted to travel into Australia and said he doubted he would get the same treatment.
Asked about the situation in an ATP Cup news conference, Jamie Murray said: "I don't know what to say about that really. I think if it was me that wasn't vaccinated, I wouldn't be getting an exemption.
"But well done to him for getting cleared to come to Australia and compete."
Djokovic may wish to elaborate further on the factors behind him being given the green light to travel into the state of Victoria, and will doubtless face media questions about the situation once he arrives.
However, the 20-time grand slam winner has been unwilling to disclose his vaccination status until now, which suggests there is little prospect of him offering further medical insight at this stage.
Asked if he felt it was not a fair decision to exempt Djokovic, Jamie Murray added: "Whatever you want to say, that's the situation."
Murray, 35, was interrupted at that point by Great Britain captain Liam Broady, who said: "At the end of the day, you have to trust he has a valid reason for the medical exemption, that's all you can say about it really, isn't it."
However, Broady had no doubt Djokovic would play the Australian Open, which begins on January 17, and claimed it was a foregone conclusion.
In an Instagram story, Broady posted a screenshot of Djokovic's announcement, adding the remark: "The second AO announced there would be exemptions 8 weeks ago we all knew."
Kyrgios beat Jan-Lennard Struff 6-4 7-6 (7-4) in Brisbane, sending down 20 aces after his pledge to give 200 Australian dollars to the bushfires effort for each service winner he hits in January.
Alex de Minaur – Kyrgios' team-mate at the event, who impressively came from a set down to defeat an ill-tempered Alex Zverev 4-6 7-6 (7-3) 6-2 – followed suit after Kyrgios tweeted his intention to help, while stars from other sports have also joined in.
Brisbane Heat captain Chris Lynn struck three sixes as he top-scored in his team's win over Hobart Hurricanes in the Big Bash League on Friday, with each maximum worth $250 to the recovery effort.
"I don't really care about the praise too much. I just think we've got the ability and platform to do something like that," Kyrgios told Amazon Prime when asked about the movement he inadvertently started.
"My home town is Canberra and we've got the most toxic air in the world at the moment, so it is pretty sad. It's tough."
Having appeared choked up at that point, Kyrgios added: "I just chucked up a tweet and everyone got behind it. It is bigger than tennis.
"It's going to all the families, firefighters, animals, everyone who is losing homes, losing families. It's a real thing.
Australia completed a 3-0 Group F win after Chris Guccione and John Peers beat German doubles pair Kevin Krawietz and Andreas Mies 6-3 6-4.
In the section's other match, Canada were similarly emphatic against Greece, with Denis Shapovalov winning a pair of tie-breaks to best Stefanos Tsitsipas after Felix Auger-Aliassime demolished Michail Pervolarakis 6-1 6-3.
Auger-Aliassime and Shapovalov then combined to win the doubles rubber against the same opponents, prevailing 6-2 6-3.
Great Britain and the United States both allowed leads to slip against Bulgaria and Norway respectively.
In Group C, Cameron Norrie beat world number 423 Dimitar Kuzmanov in three sets and Dan Evans made a fast start against Grigor Dimitrov to go a set up.
But Dimitrov prevailed 2-6 6-4 6-1 and he and Alexandar Lazarov triumphed after three tie-breaks against Jamie Murray and Joe Salisbury in the small hours of the Sydney morning.
Taylor Fritz beat Viktor Durasovic 6-2 6-2 and John Isner had the USA in charge of the second singles rubber when he took the opening tie break versus Casper Ruud, only for two match points and a second breaker to go against him. He eventually lost, going down 7-5 in the deciding set.
Ruud and Durasovic then recovered from dropping the first set to beat Austin Krajicek and Rajeev Ram 10-5 in a match tie-break in Perth.
Steve Darcis and David Goffin inspired Belgium to a 3-0 Group C win over Moldova, while Daniil Medvedev's 1-6 6-1-6-3 victory against Fabio Fognini helped clinch a 3-0 victory for Russia against Italy in Group D.
Andy Murray revealed he may play alongside brother Jamie in the men's doubles at Wimbledon, with the pair to make a decision "in the next few days".
Murray and Dan Evans fell at the first hurdle of the men's doubles at the French Open on Friday, as the British duo were defeated 7-6 (8-6) and 7-6 (7-3) by Thiago Seyboth Wild and Sebastian Baez.
It was expected to be the three-time grand slam singles champion's final appearance at Roland-Garros, after he revealed in February that he does not "plan on playing much past this summer".
Murray, who was beaten by Stan Wawrinka in the opening round of the men's singles, now turns his attention to the grass-court season, which "may" include a familiar partnership at Wimbledon with brother Jamie - a two-time mixed doubles champion at SW19.
The brothers have also played together at two Olympics - in 2008 and 2016 - while their prolific partnership helped inspire Great Britain to Davis Cup glory in 2015.
"My brother doesn't have a partner for Wimbledon currently," said the 37-year-old, who last played in the men's doubles at SW19 five years ago alongside Pierre-Hugues Herbert.
"We have spoken a little bit about it. So I may do that, but not 100 per cent sure yet. [We will decide] ahead of time. Obviously, Jamie could also get a good partner, as well. We'll see what happens, but we'll probably decide in the next few days."
Murray also reflected on his and Evans' first-round exit at Roland-Garros, where he felt his progress was hampered by an ongoing back problem.
"It's been frustrating for everyone," he added. "I felt like we had a good chance to do well, and we agreed if we were going to play, we would play to try and win the tournament.
"I have been struggling a little bit with my back the last couple of weeks, so I probably needed a few lighter days anyway."
Murray and brother Jamie suffered a 7-6 (8-6) 6-4 defeat to Australian pair John Peers and Rinky Hijikata, two days after he announced he would not participate in the singles.
The two-time Wimbledon champion, who underwent back surgery just over a week ago, has since announced he will pair up with Emma Raducanu in the mixed doubles.
Following his doubles match on Thursday, the likes of John McEnroe and Novak Djokovic were invited onto Centre Court for an emotional ceremony to mark his achievements at the tournament, which he won in 2013 and 2016.
In an interview with BBC Sport's Sue Barker, Murray said of teaming up with his elder brother: "It was obviously really special, I never had the chance to do it before.
"The way things worked out, there was a chance this year and it was a race against time.
"Physically it wasn't easy but I'm glad we could get out here and do it just one time together."
Murray has repeatedly said he will retire at the end of 2024, with an appearance at the Paris Olympics now his target.
Speaking about his impending retirement, Murray said: "It's hard. I would love to keep playing but I physically can't, it's too tough now.
"The injuries have added up and they haven't been insignificant. I want to play forever. I love the sport, it's given me so much and taught me so many lessons."
Reflecting on his two victories at SW19, Murray said the pressure of winning his first grand slam title meant he could not fully enjoy the 2013 success, but opened up on the big celebrations that followed his second triumph.
"I didn't really enjoy it as much as I should have done, I just found it very, very stressful," Murray said. "2016 was different.
"I felt way less pressure and the enjoyment I got out of that win was amazing, I could enjoy it with the people closest to me.
"Of the slams, that was my favourite one. I don't remember much of that night, I had a few drinks and I did unfortunately vomit in the cab on the way home!"
Leading male players from Britain, including Murray's brother, three-time grand slam singles champion Andy Murray, took part in a tournament at the National Tennis Centre in Roehampton last month.
Jamie Murray organised that behind-closed-doors event, where British number one Dan Evans secured the top prize, seeing off Kyle Edmund in the singles final.
The all-British competition will now return in a new format, as seven-time doubles grand slam winner Jamie Murray again takes the role of tournament director, with the week-long competition starting on July 27.
The tournament will feature two teams – each made up of six men and six women – with eight matches, including singles, doubles and mixed, taking place each day.
"It is hugely exciting to be bringing Battle of the Brits Team Tennis to the British fans at the end of July," Jamie Murray said on the Lawn Tennis Association website.
"Battle of the Brits Team Tennis will be showcasing the best of British tennis in a unique team competition. We will continue to raise funds for charity during the week of competition."
The WTA Tour is due to resume on August 3 in Palermo, Italy, with the men's ATP Tour set to start up again with the Citi Open in Washington from August 14.
The first grand slam since the coronavirus caused tennis to be suspended in March is set to take place in New York, with the US Open due to begin on August 31.
In his first competitive action since appearing at the Davis Cup Finals in November, the three-time grand slam winner recorded a 6-2 6-2 win over Liam Broady on Tuesday.
Murray admitted beforehand that he had little chance to practice before taking part in the exhibition tournament at the National Tennis Centre in Roehampton, stating before his opener that his main aim was to "get through" it with his body holding up.
The Scot has been troubled with a pelvic injury but, with his serve working well and showing the occasional glimpse of his undoubted quality, he proved far too strong for Broady.
Still, he refused to get too carried away with his performance, describing it as "okay" during his post-match interview with Amazon Prime Video.
"I served pretty well, I thought I served well throughout the match. A lot of free points there – I've been working on my serve quite a lot," Murray said.
"I didn't hit the ball that well from the back of the court, quite a lot of errors and my balance didn't feel great. I wasn't timing the ball very well, but it was alright.
"For a first match in seven months, I've not been practising much and not even doing that well in practice matches, it was alright."
However, Murray is expecting to have problems on Wednesday when he attempts to deal with Edmund, who began his campaign by beating James Ward.
"He's fit, hitting a big ball, so I'd be surprised if I manage to come through that one," the world number 129 said of his next opponent.
"If I serve like I did today and hit the ball better a little bit cleaner from the back of the court, I'll give myself some chances. But it will be tough."
The event, organised by Jamie Murray, is following strict health guidelines as it is taking place amid the coronavirus pandemic.
The five-time All England Club singles champion and six-time women's doubles winner turned 42 years old a fortnight ago, and this year marks 25 years since her Wimbledon singles debut.
Williams had been inactive on tour since last August's Chicago Open, with many doubting she would play again, but the American great showed flashes of brilliance alongside British doubles expert Murray in a 6-3 6-7 (3-7) 6-3 victory over Michael Venus of New Zealand and Poland's Alicja Rosolska.
Some 24 years on from the season when she and Justin Gimelstob landed the Australian Open and French Open mixed doubles titles, Williams thrilled the Court One crowd with her energetic play at times, showing only a hint of rust.
The match was not initially allocated a specific court, as organisers hoped play on a show court would end early to allow for it to be added to the programme.
That panned out ideally, meaning a busy stadium crowd got to see Williams and five-time grand slam mixed doubles champion Murray pair up for a late-evening tussle, three years after their siblings Serena Williams and Andy Murray also joined forces at Wimbledon.
The 42-year-old partnered Murray on Friday and rolled back the years with a 6-3 6-7 (7-3) 6-3 victory against Michael Venus and Alicja Rosolska in the first round.
That match marked Williams' first competitive action since last August's Chicago Open, with many questioning whether she would ever return to the court.
After showing some flashes of brilliance on Court One, Williams later revealed sibling Serena played a part in her decision to participate in this year's event at SW19.
Serena had herself returned from a year on the sidelines earlier in the week in the women's singles, only to go down to Harmony Tan in a three-set thriller.
"It was definitely super last minute. I was just inspired by Serena," Venus said. "It was amazing. I just was so happy to have so much help today.
"I've been trying to play with [Jamie] forever. He plays hard to get!"
All-time great Williams, who made her Wimbledon debut 25 years ago, is a five-time All England Club singles champion and has won the women's doubles on six occasions.
Williams and Murray will now face British wild cards Alicia Barnett and Jonny O'Mara in the second round, and the American says the fire is back in her belly.
"I had no plan to play but I saw the grass and I got excited," she said. "That's why I was asking [Jamie] last minute. He just had a baby, too, so I know there's a lot going on.
"I couldn't have guessed that I would be here right now, taking it at the last minute. I haven't played in a year, so you don't know what you're going to get.
"Practice is so much different from a match. It's not easy physically or mentally or anything. Just at the last it was like, 'Oh my God, wow.'
"I just not only played a match but won a match. I'm never like that kind of player. I always expect to win.
"When I sat there, we wanted to win, but when I sat there at the end, it was real. Yeah, I felt something in my heart."
The 42-year-old has not played on the WTA Tour since a first-round defeat at the Chicago Open last year due to a leg injury.
However, despite not entering either the women's singles or women's doubles in SW19, Williams was pictured arriving at the championships with a sports bag on Sunday.
And it was confirmed on Wednesday she would be teaming up with Murray, with the pair facing a first-round clash against Alicja Rosolska and Michael Venus on Friday.
Both Williams and Murray have enjoyed mixed doubles success previously at Wimbledon.
Murray is a two-time winner having partnered Jelena Jankovic in 2007 and Martina Hingis in 2017 – beating Michael Venus in the latter final – while Williams reached the final in 2006 alongside Bob Bryan. She has won the women's doubles with sister Serena Williams six times.
This is not the first All England Club partnership between the two hugely successful tennis families either, with Serena and Andy Murray reaching the third round at Wimbledon in 2019.
Last year, Venus bowed out at the second round in the singles following defeat to Ons Jabeur but vowed she would return to SW19 – a promise she has now fulfilled.
She follows the return of her sister Serena, who suffered a surprise singles defeat in the opening round against Harmony Tan on day two of the tournament.
