The victor of the quarter-final will face either Belgium or Italy in September's semi-finals.
Thompson's career accelerated past year when he won four Challenger titles, and also managed his first win in an ATP main-draw match, in Sydney.
After the match, Kyrgios was thrilled, and said that it was good to even the score against his American rival who has beaten him twice before in the past.
A win in Saturdays doubles rubber could set up a semifinal date with Italy or Belgium in September for Australia, with John Peers and Sam Groth due to take on Sam Querrey and Steve Johnson.
Thompsons victory continues the 22-year-olds undefeated run for Australia in Davis Cup after he won both his singles rubbers on debut against Czech Republic at Kooyong in February in straight sets.
Jack Sock and Steve Johnson battled past Sam Groth and John Peers to keep the USA alive on Saturday in Brisbane.
"He does a lot of things for a lot of juniors as well, coming up, so there's a lot of stuff that doesn't really get spoken about Nick Kyrgios off-court, either".
The American No.2, who owns a Davis Cup win over Roger Federer, clouted the most aces on the ATP Tour a year ago, 1159, at an average of 23 per match, the majority of which were best-of-three-set encounters.
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United States captain Jim Courier said his team was capable of winning both matches Sunday and reaching the semi-finals.
Thompson, 22 and from Sydney, couldn't help but remind fans of his captain, Lleyton Hewitt, as he scrapped and clawed through four sets against Sock, who lost 6-3 3-6 7-6 (4) 6-4 in a disappointing result for U.S. captain Jim Courier.
Two double-faults and two mistimed forehands from Sock gave Thompson the first break of the match at 5-3 in the first set.
It will be the 47th installment of the most-played tie in Davis Cup history between the great rivals, with the Americans leading 26-20. I'll still challenge the umpire, I'll still say what I think, I'll still throw my racket, I'll still swear, but I'll compete every point.
In the first match, Jack Sock of the United States took on the 79 ranked Jordan Thompson, and the Nebraska native struggled to find consistency on his serve throughout the match.
Sock's service game at 1-2 was the only game in the first set to go against the server, but two missed volleys from Johnson in it contributed.
The American rallied to win the second set, then recovered from 4-1 down to level the third set before Thompson won in a tiebreaker.