Two-time Grand Slam finalist Anderson and Sock, who has never made it past the fourth round of a major in singles, put the legends to the sword with a 6-7 (7-5) 6-3 10-6 win.
But their opponents soon found their rhythm, taking the second set 6-3, with Sock's experience as a three-time doubles major champion shining through on some impressive volley shots and Anderson displaying a strong serve.
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.
The format calls for shorter matches.
Three points per match are on offer tomorrow, with the first team to 13 points lifting the trophy.
Though Sock has turned into one of the best doubles players in the game, winning both Wimbledon and the US Open with Mike Bryan, he's still set on getting more out of his singles career.
"It's pretty nice to play indoors - no sun, no wind", Dimitrov said.
Despite the defeat, which they suffered in the decisive match tie-breaker, Team Europe finished the first day with a 3-1 lead over Team World.
Today's matches were worth two points each. "It's a nice feeling and I think it's great for you guys to see that".
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".
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.