The Capitals were awarded a man-advantage 7:09 into overtime when Ryan McDonagh was whistled for shooting the puck over the glass, causing a delay of game.
Mike Ribeiro found Green alone in the high slot 51 seconds later, and the Capitals' defenseman one-timed toward the net, where it deflected off the Rangers' Derek Stepan and into the goal.
Braden Holtby turned aside all 24 shot he saw for a shutout as Washington followed up Thursday's 3-1 victory over New York with another triumph to grab a 2-0 advantage the Eastern Conference quarterfinal series.
Henrik Lundqvist made 37 saves for the Rangers, who will host Game 3 Monday.
Celebrity Families of 2014 [PHOTOS]