Kyrgios can reserve his best for the greats of the game and has a 1-3 record against Federer.
"We've had some really close matches that haven't gone our way and I really wanted to put us back in there", said Anderson.
South Africa's Anderson got the fight back started by upsetting reigning Wimbledon and US Open champ Djokovic 7-6, 5-7, 10-6 to earn two points for Team World at the United Center arena. "I think today that's what I did". Federer played in the first Laver Cup but Djokovic is taking part for the first time.
Team Europe now lead 3-1 with two points up for grabs on day two.
That prompted vocal protests from Kyrgios, who branded the decision a "shit judgement", while Team World captain John McEnroe tongue-in-cheek commented he hoped the umpire would "get a little present after the match".
He only found his rhythm more in the second set to win 22 more winners than Kyrgios and served out the match with an eighth ace.
But amazingly their combined powers were not enough to triumph, with the pair eventually going on to lose the match up 7-6, 3-6, 6-10, to Sock and Anderson.
Under the scoring system for the Laver Cup, victories on day two are worth two points each, with victories on day three rising to three points. If the players split sets, they go straight to a 10-point tiebreaker.
Excitement had been building ever since it was announced the former world No 1s would pair up for Team Europe in the competition's second year - with the blue side hoping to take a clean sweep into the weekend after a trio of singles wins earlier in the day.
Zverev then raced into the lead in the tiebreak before closing out for the win.