The Redskins built a 28-3 halftime lead and then survived an assault by Dallas quarterback Tony Romo over the final two quarters.
Both teams have 5-6 records and will fall two games behind the NFC East leading New York Giants if the Giants can defeat Green Bay on Monday night.
Griffin completed 20-of-28 passes for 311 yards in his introduction to the Washington-Dallas rivalry. The teams will meet again on the final weekend of the season.
During the second period Griffin delivered touchdown passes of 68 yards to Aldrick Robinson, 59 yards to Pierre Garcon and 6 yards to Santana Moss. Alfred Morris also produced a 1-yard scoring run during Washington's 28-point second period.
Griffin's final touchdown throw traveled 29 yards to a wide open Niles Paul early in the fourth quarter to give Washington a 35-13 cushion.
Romo, who hooked up with Dez Bryant with an 85-yard touchdown pass in the third period, also hit Felix Jones from 10 yards out and Bryant with an 11-yarder to bring the Cowboys within seven with 8:18 to play.
Washington, however, fought off the Dallas comeback by driving 50 yards in 11 plays and getting a game-clinching 48-yard field goal from Kai Forbath with 2:58 to play.
Romo, with his running game depleted by injuries and his offensive line struggling, threw 50 passes last week in a comeback win over Cleveland. He threw 62 times against the Redskins, completing 37 of them for 441 yards with two interceptions.
This was the seventh time Washington had met Dallas on Thanksgiving and it was the first time the Redskins had won.