Bergeron's unassisted tally was his first of the season since returning this week from reconstructive knee surgery. It handed the Southeast Division-leading Lightning a victory and put them five points ahead of second-place Washington.
Dwayne Roloson turned aside 31-of-34 shots but allowed goals to Carolina's Jeff Skinner and Erik Cole 19 seconds apart in the final minute of regulation to force overtime.
Eric Staal also scored and Cam Ward made 37 saves for the Hurricanes.
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