The Sabres were trailing 3-2 with 2:01 left in regulation when Marcus Foligno put a long shot past Islanders goaltender Evgeni Nabokov, who was partially screened on the play.
Miller (41 saves) then secured the Sabres' initial victory with a final save on John Tavares.
Vanek also scored in regulation for Buffalo, which improved to 0-6-1.
Nielsen, Moulson and Matt Martin each scored and Nabokov finished with 31 saves for the Islanders.
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