The two teams combined for seven goals in the final 20 minutes. The Hurricanes entered the frame leading 2-1 and eventually came away with only their third win in seven games.
Sutter's first tally, 1:49 into the third, put the Hurricanes up 3-1. His second tied the game 3-3 after Toronto's Tim Brent had notched a pair of quick goals in between.
But the eventual game-winner went to Jeff Skinner, whose power-play tally 11 minutes into the period put Carolina ahead for good at 4-3.
Joe Corvo, Jamie McBain and Erik Cole scored and Cam Ward stopped 26-of-30 shots for Carolina.
Brent added an assist and Mike Brown notched a goal and a helper as the Maple Leafs dropped their fifth game in six tries.
Jean-Sebastien Giguere finished with 22 saves for Toronto.
Swim Week Miami Beach 2014 [PHOTOS]