Rookie running back Alfred Morris rushed for 200 yards on 33 carries and three touchdowns, while quarterback Robert Griffin III added a 10-yard TD run to lead the Redskins.
Washington (10-6), winners of seven straight, will host the Seattle Seahawks (11-5) in the first round of the playoffs next Sunday at FedExField.
The Cowboys (8-8), who would have won the NFC East and the final playoff berth by beating Washington, closed the season with two consecutive losses and missed the playoffs for the third straight year.
Griffin, the Redskins' Pro Bowl-bound rookie, completed only nine of 18 passes for 100 yards but Washington enjoyed a 274-100 advantage in the ground game.
Morris gave the Redskins a 21-10 fourth-quarter lead with a 32-yard touchdown run, reversing field against the Cowboys' pursuit. The score, with 10:32 left in the game, put the crowd in a party atmosphere not seen at FedExField in a long time.
The revelry was short-lived, however, when the Cowboys cut the lead to 21-18 on a 10-yard touchdown pass from Tony Romo to Kevin Ogletree and a successful two-point conversion toss from Romo to Dwayne Harris.
With three minutes left and Dallas ready to mount a final drive, linebacker Rob Jackson intercepted a Romo pass in the flat intended for DeMarco Murray to seal the Redskins' triumph.
A roughing-the-passer penalty by Jason Hatcher after an unsuccessful third-and-six pass by Griffin gave the Redskins a first and goal at the six.
Morris scored his third touchdown of the night, a one-yard dive, with 1:09 left to put an exclamation point on the Washington victory.
Romo completed 20 of 37 passes for 218 yards and two touchdowns but the 10-year NFL veteran had three interceptions.
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