In the singles matches, Federer had an easy 6-3 6-2 win over Kyrgios but Germany's Zverev had to save a match point against American John Isner before winning 3-6 7-6 (7-6) 10-7.
This is the second instalment of the Laver Cup which has been dubbed the Ryder Cup of tennis - although it lacks the tradition of golf's premier team event.
Novak Djokovic is a fun man and he is known for keeping it simple and enjoying his time on the court and on Saturday, he pulled off the flawless "fun" moment during a Laver Cup tie in Chicago.
"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.
At a crucial juncture in the second, with Kyrgios trying to save a break point at 1-3 with Federer holding an advantage, a line judge called his ace out.
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".
Zverev, 21, dug deep to hold off the big-serving Isner to win 3-6, 7-6 (8/6), 10-7. If the players split sets, they go straight to a 10-point tiebreaker.
The doubles game followed three singles matches earlier in the day - Bulgarian Grigor Dimitrov putting the first point on the board for Europe after beating Frances Tiafoe 6-1 6-4.
Team Europe had taken control of the match 7-1 after singles wins for Alexander Zverev and Roger Federer.
Team World remain just two points behind their European rivals after a clean sweep in the night session of the Laver Cup made it 7-5 in Chicago.
The second set was a similarly cagey affair, going with serve for the first ten games - then Sock broke Edmund, who was playing his first match since his US Open defeat to Paolo Lorenzi, and held his own to take it 7-5 and force a third-set tie-break.