The Devils remained winless in its five games this season, although three of their losses have come in overtime. Buffalo is the only other team in the NHL without a victory.
New Jersey owned a 2-1 lead when Mark Giordano scored on the power play with only 19 seconds to play in the middle period.
Monahan scored the game-winner by tipping in a shot by Sven Baertschi. Calgary has not suffered a regulation loss this year, having lost twice in overtime.
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