Toronto forced overtime when Mats Sundin scored with just 12 seconds left in the third period, but the Hurricanes came back to win their fourth game in five outings.
Vesa Toskala kept the Maple Leafs in the game by making 35 saves, but they dropped their third straight.
The game-winning goal came on the power play after Nik Antropov was called for hooking with 2:26 remaining in the extra period.
Brind'Amour charged toward the net, took a pass from Ray Whitney and fired a one-timer past Toskala.
2014: The Year in Music [PHOTOS]