"We need some matches to get going, but today I was very clear in my game plan and I got wonderful support from my team", Federer said in his post-match court-side interview.
Team Europe had taken a big lead, but now that's been cut short to 7-5.
The South African served his way out of trouble, while errors from Djokovic meant he again failed to put any points on the board after yesterday's doubles loss with Federer.
Team Europe had moved into a 7-1 lead after singles wins for Germany's Alexander Zverev and Switzerland's Roger Federer at the United Center, but Anderson's triumph and a doubles victory for the United States' Jack Sock and Australia's Nick Kyrgios reduced the deficit.
Federer and Djokovic took the first set 7-6 in a tiebreaker, but Sock and Anderson won the second set 6-3 and then captured the deciding tiebreaker 10-6 to give Team World its lone win on the first day of the exhibition.
"Even though we're the home team, I'm sure they will have a lot of fans cheering for them while they are playing together", Sock said.More news: Jacinda Ardern and baby Neve win hearts in NY
Kyrgios went down in the game and lost in straight sets against Federer, who was a superior player throughout. If points are 12-all on Day 3, a doubles set with a tiebreaker will decide the Laver Cup champion.
The Laver Cup is a three-day indoor hard court men's tennis tournament between Team Europe and Team World.
Under the format being used for the tournament, a 10-point tiebreak is used instead of a decisive third set. "When you get in a good rhythm you just want to follow it".
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. A one-set doubles match will determine the victor if points are level on Sunday.
Each player will play at least one singles match during the first two days.
On Saturday, wins will be worth two points each instead of one.