The victory was the first of the season for the embattled Giants (1-2), who endured loud grumbling from New York fans and media about the chances their season would be over should they lose on the road Sunday.
Droughns carried the ball only three times for all of three yards, while Derrick Ward gained 94 yards on 26 carries.
Eli Manning completed 21-of-36 passes for 232 yards and one TD.
The Giants had to rally from a 17-3 halftime deficit, after Washington (2-1) scored in the first quarter on a one-yard TD run by Clinton Portis and an eight-yard touchdown pass from Jason Campbell to Chris Cooley in the second.
The Redskins were shut out in the second half, while Droughns scored twice and Manning closed out the scoring with a 33-yard TD pass to Plaxico Burress.
Portis led the Redskins with 60 yards rushing on 14 carries. Santana Moss had 82 yards in the air.
Campbell finished the day 16-of-34 passing for 190 yards.