Cousins found Pierre Garcon in the end zone with 29 seconds left in regulation and then ran in a score-tying 2-point conversion, forcing an extra frame.
Kai Forbath's field goal of 34 yards following a 39-yard punt return from Richard Crawford provided the difference as the Redskins (7-6) notched their fourth consecutive win.
Griffin finished with 246 yards and a score on 15-of-26 passing before leaving the contest with an injury.
He was later diagnosed with a sprained right knee, suffered when he took a shot from Ravens defensive tackle Ma'ake Kemoeatu while scrambling late in the fourth quarter.
His status for next week was undetermined, the team said.
Alfred Morris led the Redskins' rushing corps with 122 yards and a touchdown on 23 carries.
Joe Flacco went 16-of-21 for 182 passing yards with one interception for the Ravens (9-4), who dropped a second straight and missed a chance to lock up a playoff spot.