Hagelin, a sixth-round draft choice playing in his eighth NHL game, scored a short-handed goal 11:34 into the game to break a scoreless tie, and he made it 3-1 New York with 3 minutes gone in the final frame.
Biron turned aside all 12 shots he saw in the third period en route to the victory.
Thomas Vanek scored and Jhonas Enroth made 29 saves for the Sabres.
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