The world number 16 roared in delight and was mobbed by his team mates after sealing the victory with an ace after just over two hours on Pat Rafter Arena to set up a last four meeting with Belgium or Italy.
"I couldn't be prouder of my boys," said Australia captain Lleyton Hewitt.
"They've put so much effort into this campaign and they deserve this."
Australia led 2-0 after Friday's opening singles with world number 79 Jordan Thompson stunning Jack Sock 6-3 3-6 7-6(4) 6-4 and Kyrgios easing past John Isner 7-5 7-6(5) 7-6(5).
Sock replaced Querrey in Saturday's doubles and paired up with Steve Johnson to beat Sam Groth and John Peers 3-6 6-3 6-2 2-6 6-3 to keep the visitors alive at 2-1.
Visiting captain Jim Courier continued to mix things up on Sunday and sent Querrey in for world number 15 Sock, a gambit that initially looked like succeeding in unsettling Kyrgios.
The 21-year-old Australian, who has been in the form of his life over the last month, gave up an early service break on the back of a couple of double faults.
A whipped forehand return got the set back on serve at 4-3 but Kyrgios needed an ace to save a set point and force the tiebreak.
Roared on by a partisan crowd, Kyrgios won the tiebreak 7-4 when Querrey went long with a return and 33 minutes later was two sets up after the American erred in almost identical fashion.
Querrey rustled up two break points on the Australian's first service game in the third set and although Kyrgios saved them both, the American converted a third two games later for a 3-1 lead.
Kyrgios, whose 2016 season ended with an ATP ban for not trying at the Shanghai Masters, rallied to win five of the next six games, sealing the victory with a thunderbolt of an ace - his 21st of the contest.
"I knew that I hit a bit of a flat patch at the start of the third set but Lleyton told me to compete for every point and it's easy to get up out here," Kyrgios said.
Australia have not won the Davis Cup since they bagged their 28th crown in 2003. Belgium led Italy 2-1 going into Sunday's reverse singles matches in Charleroi.
(Reporting by Nick Mulvenney; Editing by Peter Rutherford)
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