The Devils trailed 4-2 when Marek Zidlicky began a rally with his second goal of the game 9:21 into the final period. Dainius Zubrus got equalizer 2 1/2 minutes later to forge a 4-4 tie.
Jaromir Jagr launched a shot in overtime that rebounded high off Capitals goalie Braden Holtby and onto the body of Greene, who used the shaft of his stick to direct it into the net for the game-winner.
A video review upheld the officials' ruling that Greene did not use his hand to put in the puck.
Greene and Jagr each finished with a goal and two assists and Zubrus notched an assist for the Devils, who won for the third time in four games.
Martin Brodeur stopped 18-of-22 shots.
Jason Chimera and Mikhail Grabovsky scored and Holtby finished with 32 saves.