Lundqvist turned aside all 13 shots he saw in the second and third periods as well as three more in the scoreless overtime, keeping the Rangers in the game.
Miller netted the game-winner in the third round of the tiebreaker after initially losing the puck, regaining control and flipping it over Dan Ellis from a sharp angle.
Derek Stepan scored in regulation for New York, which came into the contest trailing the Hurricanes by three points for the eighth and final Eastern Conference playoff slot.
Eric Staal scored and Ellis finished with 35 saves for Carolina.
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