Green tallied on a power-play with 5:48 remaining in the third period while Alex Ovechkin and Nicklas Backstrom also found the net for the Capitals, who drew even in their Eastern Conference series at two wins apiece.
Seventh-seeded Washington rebounded after Wednesday's triple-overtime Game 3 loss to the Rangers and remained without consecutive setbacks since March 22-23.
Braden Holtby earned the victory with 18 saves.
The top-seeded Rangers lost despite 23 stops from Henrik Lundqvist.
Artem Anisimov notched a goal and an assist while Marian Gaborik also scored for New York, which will host Monday's Game 5.