Gudbranson, Mike Santorelli and Scott Timmins scored against Carolina starter Cam Ward on three straight shots in a span of 1 minute, 54 seconds in the first period. That put Florida ahead 3-0 less than 6 minutes into the game.
Nathan Paetsch also scored and Joe Callahan and Evgeny Dadonov had two assists apiece for the Panthers.
Anton Babchuk scored a short-handed goal for the Hurricanes.
Ward made 17 saves in 32 minutes before giving way to Justin Pogge, who made three stops the rest of the way. Florida's Scott Clemmensen stopped 16 shots in 32 minutes and was relieved by Jacob Markstrom, who made the final 17 saves.