The teams combined for five goals in 8 1/2-minute stretch of the final period, including a tying score by New York's Anders Lee with 2:40 left in regulation.
In the extra frame, Visnovsky intercepted a Maple Leafs clearing attempt and fired it home for the game-winner.
Michael Grabner scored a pair of short-handed goals as the Islanders broke a two-game slide in their first appearance following the Olympic break.
Evgeni Nabokov earned the win with 18 saves.
Jonathan Bernier was tagged for five goals on 35 shots.
2014: The Year in Music [PHOTOS]