Green's sixth goal this season broke a 3-3 tie 76 seconds into the extra 5-minute period.
Tim Gleason's fifth goal (shorthanded, unassisted) at 13:30 of the third had created the tie.
Capitals forward Mike Knuble returned following a 12-game absence with a broken finger.
Florida's Jiri Tlusty and Eric Staal scored in the first period, and Cam Ward was beaten four times on 40 shots for the Hurricanes, whose overall road record sank to 1-11-4.
Carolina has lost six of its last eight contests in Washington.
2014: The Year in Music [PHOTOS]