Washington held a 5-1 advantage late in the second period but saw the lead evaporate when Martin St. Louis scored twice and Richard Panik and Teddy Purcell each added third-period goals to force overtime.
A slashing call to Vincent Lecavalier in the extra session, however, put Washington on a power play and the Capitals capitalized when Mike Ribeiro sent the puck to the high slot where Green fired a shot past Mathieu Garon.
Braden Holtby turned aside 35-of-40 shots to earn the win.
Panik finished with two goals and an assist and Tom Pyatt had three assists for the Bolts.
Ben Bishop started in goal but was chased after yielding three goals on 20 shots.
Garon made 15 saves.