World number two Djokovic, who has struggled this year and missed the Miami Open with an elbow injury, produced a solid performance to beat Spain's Albert Ramos-Vinolas 6-3 6-4 6-2 in Belgrade as Serbia also moved 2-0 ahead in their tie.
A below-strength France won both opening-day games against Andy Murray-less Britain in Rouen while Steve Darcis inspired Belgium into a 2-0 lead over Italy in Charleroi.
Australia and the U.S. are meeting for the 47th time with the Americans leading 26-20 overall.
But Lleyton Hewitt's side are in control after world number 79 Jordan Thompson pulled off a surprise 6-3 3-6 7-6 6-4 victory over a wayward Jack Sock in the opening rubber.
That paved the way for Kyrgios to ease past John Isner 7-5 7-6(5) 7-6(5) in a duel of the big servers.
"We knew it would be tough today but Jordan set the tone," the 21-year-old Kyrgios said. "He came out and played like he has played a million of these matches."
American duo Sam Querrey and Steve Johnson will have to beat Sam Groth and John Peers on Saturday to keep the tie alive.
Djokovic, who led Serbia to the title in 2010 since when his grand slam haul has risen to 12, was satisfied with his form after some question marks about the state of his game.
"I've lost my best game in the last few months, it was hard for me to accept it after being on top for the last six years," he said. "Today shows that I can get back to winning ways."
Viktor Troicki brushed aside Pablo Carreno Busta 6-3 6-4 6-3 in Friday's second singles.
Like Spain, 2015 winners Britain did not win a set in Rouen.
World number 17 Lucas Pouille, playing as France's number one in the absence of Jo-Wilfried Tsonga and Gael Monfils, beat Kyle Edmund 7-5 7-6(6) 6-3 before Jeremy Chardy crushed Dan Evans 6-2 6-3 6-3. Murray is missing because of an elbow injury.
Belgium are well on the way to booking a semi-final spot after Darcis, hero of their first-round defeat of Germany, beat Paolo Lorenzi in four sets. David Goffin then comfortably disposed of Andreas Seppi.
Elsewhere Cypus's Marcos Baghdatis saw his record 36-match winning run in Davis Cup singles ended by Tunisia's Moez Echargui in a Euro Africa Zone Group II tie.
(Reporting by Martyn Herman, editing by Neil Robinson)
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