The Redskins improved to 6-6 with their third straight victory, tied with the Dallas Cowboys and on the heels of the Giants, who have lost three of four to fall to 7-5.
Griffin completed 13 of 21 passes for 163 yards and ran five times for 72 yards, breaking Cam Newton's NFL record for yards rushing by a rookie quarterback.
Griffin lost the ball on one of his runs, but it flew into the arms of teammate Joshua Morgan, who ran it in for an early touchdown.
Another fumble by the Redskins -- this one by Alfred Morris, midway through the third quarter -- was recovered by the Giants and set off a melee, before leading to one of Lawrence Tynes' three field goals.
Eli Manning threw a touchdown pass to Martellus Bennett for the Giants, but it was Griffin's pass touchdown to Pierre Garcon in the fourth quarter that put Washington ahead for good.