The Jets (7-5) were trailing 16-13 early in the final quarter, but outscored the Redskins 21-3 down the stretch to remain two games in back of first-place New England in the AFC East.
Greene gained 88 yards on 22 carries while Mark Sanchez completed 19-of-32 passes for 165 yards and a touchdown in the victory.
It was more of the same for Redskins (4-8), who dropped their seventh in eight contests.
Roy Helu led the way with 100 rushing yards on 23 times carries while also catching four passes for 42 yards.
Washington's Rex Grossman went 19-of-46 through the air for 221 yards and an interception.