The Knicks looked done for when they trailed 93-71 with 7:52 left in regulation. According to ESPN analytics, the Cavs had a 99.9% chance of winning at that point.
But Brunson spearheaded an 18-1 New York run and banked in a floater to tie it at 101-101 with 19.3 seconds left in regulation, eventually making it the largest playoff comeback in the Knicks' history.
When Cleveland's Sam Merrill missed a three-pointer, they went to overtime in which the Knicks scored the first nine points.
Donovan Mitchell scored 29 points to lead the Cavaliers, and James Harden added 15.
But the Knicks will have a chance to double their lead in the best-of-seven series when they host Game 2 on Thursday.
The winner of the series will face either the defending champion Oklahoma City Thunder or the San Antonio Spurs in the NBA Finals.
The Spurs, fueled by a 41-point performance by Victor Wembanyama, beat the Thunder in double overtime in game one of the Western Conference finals.
It's the first time in NBA history that the first two games of both conference finals have been decided in overtime.