Trailing by six points late in the third quarter, Griffin started the Redskins on a 12-play, 86-yard drive, highlighted by Alfred Morris' 16-yard run.
The rookie signal-caller hooked up with Garcon for the decisive touchdown 3 1/2 minutes into the final frame, handing Washington (6-6) the lead for good.
Griffin went 13-of-21 through the air 163 yards and no interceptions while adding 72 rushing yards on five carries as the Redskins notched a third straight win and kept their postseason hopes alive.
Garcon finished with eight catches for 108 yards while Morris racked up 124 yards on the ground on 22 carries for Washington.
Eli Manning picked up 280 aerial yards on 20-of-33 efficiency with a TD for Giants (7-5), whose lead over the second-place Redskins and Dallas Cowboys shrank to one game with the loss.
Ahmad Bradshaw ran the ball 24 times for 103 yards and Victor Cruz made five catches for 104 yards in New York's defeat.