The Knicks looked done for when they trailed 93-71 with 7:52 left in regulation.
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.
When Cleveland’s Sam Merrill missed a three-pointer they went to the extra session, in which the Knicks scored the first nine points.
Donovan Mitchell scored 29 points to lead the Cavaliers and James Harden added 15.
The Knicks will have a chance to double their lead in the best-of-seven series when they host game two on Thursday.
The winner of the series will face either the defending champions Oklahoma City Thunder or San Antonio Spurs in the championship series.
The Spurs, fuelled by a 41-point performance by Victor Wembanyama, beat the Thunder in double-overtime in game one of the Western Conference finals on Monday.
It’s the first time in NBA history that the first two games of both conference finals have been decided in overtime. (AFP)
