Bergeron's goal from the right circle came off a pass from Sean Bergenheim.
Carolina had tied it 3-3 on late goals by Jeff Skinner and Erik Cole after goalie Cam Ward went to the Hurricanes bench for an extra skater.
Vincent Lecavalier scored his fourth goal in three games, and Dwayne Roloson made 31 saves as Tampa Bay won for the eighth time in 10 games.
Tampa Bay also got goals from Steve Downie and Bergenheim. The Southeast Division-leading Lightning increased their cushion over second-place Washington to five points.