Kyrgios and De Minaur kept it together and faced down four match points to take the set 18-16.
The 2-1 victory gave host Australia the first spot in the semifinals of the new global team event.
Kyrgios claimed the dramatic victory ranked as one of the best moments of his career.
The inaugural ATP Cup is a 24-nation knockout team tournament that has been set up by the men's professional tour to kick off the 2020 season.
At Ken Rosewall Arena at Sydney's Olympic Park, it was like a soccer stadium full of noise and chanting for the home team.
The tiebreaker was one-sided, with Evans winning five consecutive points from 2-2 to ensure Britain and Australia were tied after the singles. The hosts took the lead as expected after Nick Kyrgios trounced Cameron Norrie in straight sets 6-2, 6-2 - the Aussie mixing some brilliant touch and power play against the inexperienced Brit.
Australia captain Lleyton Hewitt gambled on sending his two singles players back out for the doubles against the established British team of Jamie Murray, a victor of two major doubles titles, and Joe Salisbury.
Canadian Denis Shapovalov earlier picked up his second win against a top-10 player with a 6-2 6-2 thumping of Alexander Zverev as the German continued to struggle with his serve ahead of this month's Australian Open.
"Great. You know, I think we've got great insight to how Rafa plays", Kyrgios said. "But it was awesome".
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In a daring selection gamble, Australian captain Lleyton Hewitt replaced doubles specialists John Peers and Chris Guccione with his two singles spearheads after an inspired Dan Evans levelled the tie for Great Britain with an epic victory over de Minaur. "I enjoyed it", Evans said. We just couldn't get to the last point.
The celebrating duo embraced on the court before Kyrgios threw de Minaur on his shoulder and paraded him around the court to the delight of the jubilant Sydney crowd.
"It was unreal", said Kyrgios on the ATP Cup website.
He said he expected to get a fine for the code violation but otherwise was satisfied to win the "super tough match" 6-4, 4-6, 6-3 on Thursday to secure Russia's win over Argentina.
"I sneaked it and this is what team competition is always about, particularly Britain and Australia, it's always a great rivalry and there's been great spirit". "I'm not going to lie".
The Australians will play the victor of Friday's quarterfinal between Rafael Nadal's Spain and Belgium.
Nadal was on the practice courts adjacent to Ken Rosewall Arena practicing after Australia had won the first of the quarterfinals and Russian Federation met Argentina in the second.
Kyrgios hit virtually the same shot moments later, caressing a sweet backhand return deep into Great Britain's court to send Australia up 17-16 with the ball in Kyrgios' hand.
"Jamie Murray will rue the way the second of those match points played out".