Washington (10-6) won its seventh consecutive game to capture its first division title in 13 years. The Redskins will play host to the Seattle Seahawks next Sunday in their first playoff game since 2007 and their first home postseason contest in 13 years.
Dallas (8-8) lost a winner-take-all, Week 17 battle for the NFC East crown for the third time in five years.
The game was tied 7-7 midway through the third quarter when Morris took command.
The rookie sixth-round draft choice carried on five straight plays, breaking Clinton Portis' season team rushing record, as Washington moved from its 15-yard-line to the Dallas 38.
Griffin then took over, completing two passes to receiver Pierre Garcon for 28 yards before the rookie quarterback scooted inside the left pylon for a 10-yard touchdown that put the Redskins on top 14-7 with 3:10 left in the quarter.
The Cowboys drove from their 26 to the Washington 35 but had to settle for Dan Bailey's 48-yard field goal that drew them within 14-10 in the first minute of the fourth quarter.
Morris then completed the next series with a 32-yard touchdown burst, the biggest run of his 33-carry night.
Dwayne Harris' 39-yard punt return and Sav Rocca's facemask penalty gave Dallas the ball on the Washington 16, and Tony Romo found receiver Kevin Ogletree for a 10-yard touchdown. Romo's two-point-conversion pass to Harris made it 21-18 with 5:50 left.
Washington linebacker Rob Jackson intercepted a Romo pass at the Dallas 27 with three minutes remaining to set up Morris' third touchdown and seal Washington's renaissance.
Griffin completed nine of 18 passes for 100 yards, and he added 63 yards on six carries. Romo went 20-for-37 for 218 yards with two touchdowns and three interceptions.
Washington needed a timeout after the first play because the communications device in Griffin's helmet was malfunctioning. After a three-and-out, Harris returned Rocca's punt 28 yards to the Washington 27. However, Romo and receiver Ogletree weren't in sync on the third-down pass that Redskins rookie cornerback Richard Crawford intercepted at the 17.
Two runs by Morris for 29 yards put Washington on the Dallas 42. Griffin picked up the next first down with a 9-yard jaunt. However, the drive stalled at the 19, and kicker Kai Forbath's perfect debut season ended on his 18th field-goal attempt when he hit the right upright from 37 yards.
The Cowboys picked up their initial first down on a diving 23-yard grab by Dez Bryant on third-and-14 at their 23, but Romo's next pass was underthrown, and cornerback Josh Wilson picked it off at the Washington 13.
On Dallas' first series of the second quarter, Harris got wide open on a third-down slant and raced 25 yards to the Washington 49. The Cowboys finally broke the scoreless tie eight plays later. Romo had all day to find tight end Jason Witten in the end zone for a 9-yard touchdown on third-and-goal. Dallas led 7-0 with 7:59 left in the first half.
With Morris running four times for 45 yards, the Redskins responded with an eight-play, 68-yard drive that ended with his 17-yard sprint to the left pylon with 3:28 remaining before halftime.
NOTES: Witten's first catch made him the third tight end with 800 in his career, joining Atlanta's Tony Gonzalez and Hall of Famer Shannon Sharpe. ... Redskins reserve cornerback D.J. Johnson departed in the first quarter with a sprained left knee. ... Cowboys receiver Miles Austin exited on Dallas' first series of the second quarter with a left ankle injury. ... Morris' touchdown set a Redskins rookie record, topping the 10 that Hall of Famer Charley Taylor scored in 1964.