New York overcame a 3-1 deficit and held the Canadiens scoreless for the final 31:30 to end Montreal's five-game winning streak.
Montreal built its early lead thanks in part to two goals from Max Pacioretty, but Matt Moulson countered with a pair of scores for New York and Frans Nielsen tied it for the Islanders with the only third-period goal.
Hickey came through in overtime as he played in his ninth NHL contest.
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