DeMarco Murray rushed from four yards to put the home side on the board with just over six minutes on the clock in the first quarter, but Kai Forbath reduced that lead to four points with a 20-yard field goal.
The Cowboys extended their advantage when Dwayne Harris collected a punt and returned 86 yards for a touchdown with two minutes remaining before half time.
Forbath nailed kicks either side of the break to get his team within five points, but the home side pulled further ahead thanks to a brilliant pass from Tony Romo, and an equally good catch from Terrance Williams.
Alfred Morris rushed the middle to keep the Redskins in the game heading into the fourth quarter, and Forbath had the chance to kick from 49 yards. However, this tie he could not work the scoreboard.
Dan Bailey edged the Cowboys further ahead from 30 yards before Joseph Randle was bundled over the line by his own teammate Travis Frederick, putting the result almost beyond doubt with over eight minutes remaining.