Team World went into the final day of action at the O2 Arena in London sitting four points behind its opponent, but it produced a stirring fightback to claim the trophy at the fifth time of asking.
Central to its success was Canadian Auger-Aliassime, who beat Novak Djokovic in singles after successfully teaming up with Jack Sock in the doubles.
Holding an 8-4 lead from Sunday (AEST), many expected Team Europe to breeze it from there, but John McEnroe's World team had other ideas and earned a 13-8 victory.
Up first in the doubles were Auger-Aliassime and Sock, who lost the first set to Team Europe's Andy Murray and Matteo Berrettini.
However, the World duo roared back to deliver three points for its team by claiming a 2-6 6-3 10-8 victory.
Djokovic won two matches for Team Europe on Sunday, yet he was powerless to stop Auger-Aliassime in their singles clash. The 22-year-old Canadian landed a 6-3 7-6 (7-3) over the Wimbledon champion.
That moved Team World 10-8 ahead in the overall contest, setting the stage for a decisive clash between Stefanos Tsitsipas and Frances Tiafoe, with a further three points at stake.
Greek Tsitsipas won the first set, but 24-year-old American Tiafoe stormed back to win an epic tie-break in the second on his way to a 1-6 7-6 (13-11) 10-8 success.