Europe, which has won the prior two editions of the competition, lead 7-5 heading into the final day, but might have had one hand on the trophy already were it not for Kyrgios.
A home side able to name a line-up where each of their six players are ranked higher than John Isner, its opponent's top star, celebrated singles triumphs for Federer and Rafael Nadal but found Kyrgios in determined mood.
Team World had struck first to level the contest at 3-3, with wins on the second day worth two points, double its first-day value, while third-day victories earn three.
Isner saw opponent Alexander Zverev fall apart and clinched a 6-7 (2-7) 6-4 10-1 success.
Pegged back, matters looked to be going from bad to worse for Europe when the dangerous Kyrgios took the first set against Federer.
But the Swiss veteran recovered to triumph 6-7 (5-7) 7-5 10-7 and it appeared to be plain sailing from there, with Nadal winning the final singles contest of the day, easing past Milos Raonic 6-3 7-6 (7-1) to regain complete control of the competition.
"Being here for me is a very positive energy," Nadal said. "I like being around such a great team and the crowd is amazing.
"I feel very lucky [to win] the first set. I started to play better later in the match and I think I played a very good tie-break in the end."
The Spaniard was back on court to close out the day's play, teaming up with Stefanos Tsitsipas, to take on Kyrgios and Sock.
However, Kyrgios gave an early indication of what was to come as he started in impressive fashion again with some blistering winners in an opener Team World dominated, breaking swiftly and protecting the serve with ease.
The Australian was guilty of gifting away the decisive first break in the second set, making a mess of a seemingly straightforward volley, before coming into his own again with the match level.
Nadal was too often left to plough a lone furrow by Tsitsipas and Kyrgios excelled at the net, greeting each European error in typically extravagant fashion.
Fittingly, after a one-sided tie-break, the match was clinched on Kyrgios' serve 6-4 3-6 10-6 to set up an intriguing final day.