The pair defeated Rafael Nadal and Stefanos Tsitsipas 6-4 3-6 10-6 in Geneva to leave the overall score at 7-5 in Europe’s favor – with 13 points required for victory.
After beginning the day 3-1 down, John Isner brought World level with a singles win over Alexander Zverev.
However, Roger Federer restored Europe’s two-point lead with a pulsating win over Kyrgios, and Nadal then beat Milos Raonic to extend the gap to 7-3.
It left Team World in serious need of a result in Saturday night’s closing doubles.
They had a strong team to rely on, Sock and Kyrgios eventually proving too much for Europe’s scratch pairing as they took their Laver Cup record together to 3-0.
That’s one of the best 10-point tiebreakers I’ve ever seen a doubles team play – John McEnroe
“There was a lot on the line there,” said Sock.
“We obviously wanted to keep the weekend close and being down 9-3 is a lot different to being 7-5.”
Kyrgios added: “Jack’s had some tough injuries and to come back, put on a performance in singles and be our best doubles player is amazing.
“Our whole team is coming together and we’re going to ride this momentum into day three.”
A heavy Sock forehand winner gave World an immediate break of serve that would take them all the way to the opening set.
Federer was a leading figure on the sidelines, offering advice to Nadal and Tsitsipas at the changeovers, and the Spanish-Greek pairing at one stage had to clarify their signaling to each other.
The confusion clearly passed as Europe fought their way back into the contest with two breaks of serve in the second set, forcing a decisive 10-point match tiebreaker.
The blue side made the first move with a mini-break for 4-3 but it sparked a burst of brilliance from the reds, Kyrgios and Sock reeling off six points in a row to all but seal the win.
“They stepped up,” said Team World Captain John McEnroe.
“That’s one of the best 10-point tiebreakers I’ve ever seen a doubles team play, and they needed to because Rafa and Stefanos wanted it bad.
“It gives us some momentum going into tomorrow.”