The 22-year-old Thompson made his Davis Cup debut by defeating Czech No. 1 Jiri Vesely 6-3, 6-3, 6-4, finishing the match with an ace. Thompson was called into the team when Australia's No. 2-ranked player, Bernard Tomic, made himself unavailable for selection.
Kyrgios beat Jan Satral 6-2, 6-3, 6-2 in the second singles match on hard courts at Kooyong.
In his first match following his second-round Australian Open loss to Andreas Seppi, the 15th-ranked Kyrgios made light work of the world No. 157 Satral. Kyrgios won the first three games of each set but needed six match points to clinch it in 96 minutes.
Thompson broke No. 54 Vesely in the opening game of the match and apart from a slight stumble in the third set, was never in trouble.
"That was huge, to get off to a great start and break first game and then hold, that gave me a lot of confidence," Thompson said. "It really relaxed me and I really took it from there and sort of ran away with the match."
Thompson won the men's doubles title with fellow Australian Thanasi Kokkinakis at the Brisbane International, made the second round of the Australian Open singles draw for the first time and has reached a career-high ranking of 65.
Sam Groth and John Peers are expected to play doubles for Australia on Saturday, but the makeup of the Czech pairing was unclear.
Czech captain Jaroslav Navratil said former world No. 8 Radek Stepanek, who has had a history of back problems, was in doubt for doubles.
"It's a difficult situation for us because Australia has a very good doubles team and Radenek is not playing because yesterday (Thursday) he gets injured a little bit, so it's tough for us," Navratil said.
He said he would speak with Stepanek again on Saturday, but otherwise Satral and Vesely would play doubles for the Czechs.
Peers won the Australian Open doubles title last week with Finland's Henri Kontinen, beating Bob and Mike Bryan of the United States in the final.
The Australia-Czech winner will play either the United States or Switzerland in the quarterfinals in April — in Switzerland if the Swiss win, in Australia if the Americans advance.