Washington will host the Seattle Seahawks next Sunday, the Redskins' third consecutive playoff game against the Seahawks. They lost at Seattle in the 2005 and 2007.
The Cowboys (8-8), meanwhile, will miss the playoffs for the third straight season, having stumbled in a make-or-break end-of-regular-season game for the third time in five years.
Tony Romo threw three interceptions matching his total from the last eight games combined. A poor throw was picked by Rob Jackson when the Cowboys had a chance to drive for a winning score in the final minutes.
Romo almost became the first Dallas quarterback to throw for 5,000 yards in a season, but his career is instead further tainted by post-Christmas disappointments. He also had Week 17 losses to the Philadelphia Eagles (44-6) in 2008 and the New York Giants (31-14) last year, as well as his 1-3 record in playoff games.
Morris finished with 1,613 yards, topping Clinton Portis' 1,516 in 2005. He was especially dominant in the Redskins' go-ahead drive in the third quarter, when six plays were runs by Morris and the other three involved fake handoffs to him. The touchdown came when Griffin faked to Morris one of several times linebacker DeMarcus Ware was totally fooled by deception in the backfield and ran 10 yards around the left end. It put Washington ahead 14-7 in the third quarter.
The Cowboys answered with a field goal early in the fourth, but Morris' 32-yard scamper gave the Redskins a 21-10 cushion with 10:32 to play.
Trying to play catch-up, Dallas pulled within three on a 10-yard pass to Kevin Ogletree and a 2-point conversion with 5:50 to play. But Morris' third touchdown, a 1-yard run with 1:09 left, sealed the win.
Playing against a defense missing its five best run defenders, the Redskins didn't need Griffin to throw much. He completed just 9 of 18 passes for 100 yards.
The Redskins were calling designed runs for Griffin as a regular part of the game plan for the first time since he sprained his right knee four weeks ago. He lacked the explosiveness he showed earlier in the season, perhaps hampered by his big brace, but he was still a running threat.
Romo completed 20 of 31 passes for 218 yards.
The Cowboys also dealt with in-game injuries to receivers Miles Austin (left ankle) and Dez Bryant (back). Bryant, who had a torrid second half of the season despite breaking his left index finger, had four catches for 71 yards.
The Redskins also set a franchise record for fewest turnovers in a season with 14, fewer even than the 1982 team that played only nine regular-season games because of a players strike.
Washington's slow start this season prompted coach Mike Shanahan to dismiss playoff hopes and declare the remaining seven games would determine which players would be on his team "for years to come."
Griffin and his teammates had other plans, and the coach quickly changed his tune. Now the Redskins are in the postseason for the first time since a wild-card berth under Hall of Fame coach Joe Gibbs in 2007.
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Dallas 18, Washington 28
Dallas 0 7 0 11 18
Washington 0 7 7 14 28
Dal • Witten 9 pass from Romo (Bailey kick), 7:59.
Was • Morris 17 run (Forbath kick), 3:28.
Was • Griffin III 10 run (Forbath kick), 3:10.
Dal • FG Bailey 48, 14:09.
Was • Morris 32 run (Forbath kick), 10:32.
Dal • Ogletree 10 pass from Romo (Harris pass from Romo), 5:50.
Was • Morris 1 run (Forbath kick), 1:09.
Attendance • 82,845.
First downs 20 24
Total Net Yards 296 361
Rushes-yards 22-100 42-274
Passing 196 87
Punt Returns 3-78 3-18
Kickoff Returns 5-93 4-109
Interceptions Ret. 0-0 3-15
Comp-Att-Int 20-37-3 9-18-0
Sacked-Yards Lost 2-22 1-13
Punts 4-41.5 5-41.4
Fumbles-Lost 0-0 1-0
Penalties-yards 1-6 2-20
Time of Possession 26:34 33:26
Rushing • Dallas, Murray 17-76, F.Jones 5-24. Washington, Morris 33-200, Griffin III 6-63, Royster 2-9, Moss 1-2.
Passing • Dallas, Romo 20-37-3-218. Washington, Griffin III 9-18-0-100.
Receiving • Dallas, Witten 7-56, Bryant 4-71, Hanna 3-20, Ogletree 2-25, Harris 1-25, Vickers 1-11, Murray 1-6, F.Jones 1-4. Washington, Garcon 3-46, Moss 2-22, Morris 2-12, Hankerson 1-12, Paulsen 1-8.
Missed Field Goals • Washington, Forbath 37 (WR).