Harris' score early in the second quarter put Dallas ahead 14-6 and he went on to rack up 222 total return yards as the Cowboys (3-3) rebounded from last week's tough loss to the Denver Broncos.
Tony Romo completed 18-of-30 passes for 170 yards with an interception and a touchdown toss to Terrance Williams in the victory.
Joseph Randle and DeMarco Murray each added a rushing score for Dallas.
The Redskins (1-4) stumbled coming off a bye week as Robert Griffin III was picked off once and lost a fumble. He finished 19-of-39 for 246 passing yards.
Alfred Morris provided 81 rushing yards and a touchdown on 16 carries for Washington.