Although Team Europe went into the finale of the three-day tournament leading 8-4, Team World were still in the hunt as Sunday's matches counted for three points each.
Team Europe needed two victories from four matches on Sunday to make it a fifth successive Laver Cup triumph, while Team World had to win three contests to claim their first title in the competition.
Canadian rising star Auger-Aliassime teamed up with American Sock to secure the first of those three required victories, setting the stage for a dramatic finish to the tournament.
Two days after his emotional farewell to tennis, Roger Federer was again watching courtside alongside his Team Europe colleagues as he takes his first steps into retirement.
With Federer cheering them on, Murray and Berrettini powered through the first set with a break in the third game and another in the seventh.
But Auger-Aliassime and Sock hit back to take the second set, then held their nerve in the first to 10 points 'Laver Breaker'.
Team Europe will turn to Wimbledon champion Novak Djokovic in the second rubber of the day when the Serb faces Auger-Aliassime.
Djokovic won his first match for two months on Saturday when he defeated Team World's Frances Tiafoe.
The 21-time Grand Slam champion also joined forced with Berrettini to win a doubles match on Saturday.
In Sunday's final two games, Stefanos Tsitsipas of Team Europe plays Tiafoe before Casper Ruud takes on Team World's Taylor Fritz.
