It was another disappointing night for Djokovic, who teamed up with Roger Federer on Friday when the duo suffered a shock doubles loss to Anderson and Jack Sock.
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.
Federer played in the first Laver Cup but Djokovic is taking part for the first time.
Britain's Kyle Edmund, a Laver Cup debutant, claimed an impressive 6-4 5-7 10-6 win over Sock and match three saw Belgium's David Goffin add to Europe's advantage as he beat Argentina's Diego Schwartzman 6-4 4-6 11-9.
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.
The exhibition is taking place on USA soil for the first time and features 12 of the world's top players squaring off against each other in a three-day showcase which wraps up Sunday in Chicago.
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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.
At times, Kyrgios also was sharp, and sent blitzing passing shots past Federer but the world No.2 was the superior player and won 22 more points.
"Kevin was so well prepared for this, we needed this bad", said World captain John McEnroe.
Then Edmund broke the Sock serve in the tenth to edge to a 6-4 first-set lead.
Under the format being used for the tournament, a 10-point tiebreak is used instead of a decisive third set.
Four matches remain on Sunday, each worth three points, with doubles to be followed by three singles rubbers.