Morris scored on runs of 17, 32 and 1 yards on 33 carries, giving the rookie 1,613 yards for the season as the Redskins (10-6) claimed their first NFC East crown since 1999.
Robert Griffin III completed 9-of-18 passes for 100 yards while also running the ball six times for 63 yards and a score for Washington, which rebounded for a division title after starting the season 3-6.
Washington will face Seattle Sunday in the NFC wild card playoff round.
Dallas, meanwhile, finished at 8-8 for the second straight year and missed the post-season for the third consecutive season.
The Cowboys' Tony Romo went 20-of-37 for 218 passing yards and two touchdowns, but was also intercepted twice.