The rookie passed for 163 yards including a touchdown, but also had some good fortune when his fumble went straight to a teammate who completed a touchdown.
On second-and-2 from the New York 28, Griffin ran to his left and was being tackled by Stevie Brown after what would have been a 12-yard gain.
But as Griffin fell with his back to the ground, the ball popped out and went right on the fly to Washington receiver Joshua Morgan three yards farther upfield. Morgan then ran 13 yards for the score.
"We didn't run (that play) in practice because we wanted to save it for the game," an amused Griffin said afterwards.
"I knew he was going to be there for it. ... Joshua did a good job being in the spot where he was supposed to be."
The win improves Washington's record to 6-6, one game back on the NFC East-leading Giants.
Elsewhere, quarterback Michael Vick has lost his starting role for the Philadelphia Eagles following his injury.
Andy Reid, the Eagles' coach, said Vick would serve as backup to Nick Foles if and when he returns from a concussion.
"He was very positive about it, completely understood and was on board," said Reid.
"Where we're at right now in the season, I think it gives (Foles) an opportunity to play and finish it up."
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