Djokovic, victor of the past two grand slam titles, double-faulted on set point to give the South African the advantage.
He was named as Federer's doubles partner on day one of the competition in Chicago and the legends surprisingly succumbed to a 7-6 (7-5), 3-6, (6-10) defeat to Team World's Jack Sock and Kevin Anderson.
Earlier in the day, Federer and German Alexander Zverev sparked Europe by winning their singles matches, the Swiss rolling past Kyrgios 6-3 6-2.
The two stars, who share a great camaraderie on and off the tennis courts, seemed to enjoying playing together.
"I felt good from the beginning of the match and that's not usual", said Federer.
The moment occurred in the very first game.
Djokovic put his head in his hands as soon as his shot struck Federer in the back, and had an embarrassed look on his face.
Heading into Saturday's match, 14-time grand slam victor Djokovic held a 6-1 career advantage, including a 6-2, 6-2, 7-6 (7/3) win over Anderson in July at Wimbledon.
This is the second installment of the Laver Cup, which has been dubbed the Ryder Cup of tennis although it lacks the tradition of golf's premier team event.
Heading into Sunday afternoon's final session, Team Europe leads Team World 7-5.
Despite Federer and Djokovic's defeat, Team Europe still lead 3-1 after the first day's play in Chicago.
Europe had earlier built their advantage on singles victories for Bulgarian Grigor Dimitrov, Briton Kyle Edmund and Belgian David Goffin.
Roger Federer put on a masterclass against Nick Kyrgios at the Laver Cup.
Day Two will feature three more singles clashes, including Federer v Nick Kyrgios, and an additional doubles match, with two points on offer for any victory.