Last week, Cousins came off the bench after Griffin got hurt and rallied the Redskins to an overtime win. Cousins was behind center from the start of this one and delivered a performance that extended Washington's longest winning streak since 2007 and moved the Redskins into a tie for first in the NFC East.
Rookie Trent Richardson had a pair of TD runs for the Browns (5-9).
Alfred Morris, a childhood rival of Richardson, had two touchdowns for the Redskins.
Wearing a burgundy warmup suit instead of his No. 10 jersey, Griffin, who sustained a mild sprain in the fourth quarter last week against Baltimore, cheered from the sideline as Cousins kept the Redskins (8-6) moving toward a possible spot in the playoffs.
Cousins may not have RG3's talent, but the fourth-round pick from Michigan State was efficient, accurate and only made one major mistake an early interception to set up Cleveland's first TD.
Cousins finished 26 of 37 as the Redskins improved to 5-0 since their bye. They'll end the season with games against NFC East rivals Philadelphia and Dallas, teams they beat in consecutive weeks to start their streak.
Griffin's playing status was in doubt all week before the Redskins surprisingly announced late Saturday night that Cousins would start. If the Browns thought they were getting a break, Cousins proved to be as challenging to stop as his more elusive and hyped teammate.
The Browns (5-9) had their winning streak stopped at three and faint playoff hopes snuffed out. Coach Pat Shurmur's fate may have rested on the Browns winning out, but that won't happen and his days in Cleveland could be dwindling quickly.
Rookie quarterback Brandon Weeden struggled from the outset. He went 21 of 35 for 244 yards and two picks, both leading to Washington touchdowns.
Cousins' second TD pass to Hankerson, a 2-yarder, gave the Redskins a 24-14 lead. As Cousins came to the sideline, Griffin gave him a fist bump and then sat on the bench with his backup, who for the past two weeks has shown he too can lead the Redskins to victory.
Washington went ahead 31-14 on Evan Royster's 4-yard run.
Weeden came back with a 69-yard scoring strike to speedy rookie Travis Benjamin to pull the Browns within 10, but Cousins calmly took the Redskins down the field again with Morris scoring from eight to make 38-21.
The Redskins needed this win, and Cousins made sure they got it.
He shook off the early pick and showed exceptional play faking ability. With Griffin out and inactive, Redskins coach Mike Shanahan had to alter his game plan and Cousins ran it with precision.
Linebacker Rob Jackson's interception of Weeden helped the Redskins take a 17-14 lead less than two minutes into the third quarter on Morris' 3-yard TD run.
Jackson picked off Weeden and returned it to the 15. Three plays later, Morris pulled in for his eighth rushing TD, tying the team rookie record set by Skip Hicks in 199
Richardson's second TD run gave the Browns a 14-10 lead with 1:11 left before halftime.
Cleveland's offense had been in a funk for most of the second quarter before Weeden completed five passes to get the Browns to Washington's 3. Richardson did the rest, busting over the left side for his 11th rushing TD. Earlier, he broke Hall of Famer Jim Brown's 55-year-old team rookie record for rushing TDs.
Kai Forbath's 44-yard field goal put the Redskins ahead 10-7. Forbath has started his career 15 of 15.
Cousins threw a 54-yard TD pass to Hankerson in the first quarter to tie it 7-7.
The Redskins gained just seven yards on their first four plays before Cousins, rolling right on the kind of play Shanahan has devised to maximize Griffin's wondrous skills, hooked up with Hankerson.
Browns safety T.J. Ward intercepted Cousins, on a ball tipped by cornerback Sheldon Brown, to set up Richardson's TD run in the first quarter. Ward returned the pick to Washington's 6-yard line, where he was tackled by Cousins, who showed good form in preventing an easy TD.
On the next play, Richardson took a handoff to the left side, was stopped at about the 2 and kept his legs and the pile moving into the end zone.
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Washington 38, Cleveland 21
Washington 7 3 14 14 38
Cleveland 7 7 0 7 21
Cle • Richardson 6 run (Dawson kick), 8:41.
Was • Hankerson 54 pass from Cousins (Forbath kick), 2:16.
Was • FG Forbath 44, 5:46.
Cle • Richardson 1 run (Dawson kick), 1:11.
Was • Morris 3 run (Forbath kick), 13:14.
Was • Hankerson 2 pass from Cousins (Forbath kick), 7:40.
Was • Royster 4 run (Forbath kick), 11:19.
Cle • Benjamin 69 pass from Weeden (Dawson kick), 10:11.
Was • Morris 8 run (Forbath kick), 5:00.
Attendance • 65,083.
First downs 25 13
Total Net Yards 430 291
Rushes-yards 35-122 15-58
Passing 308 233
Punt Returns 2-18 2-15
Kickoff Returns 1-22 7-191
Interceptions Ret. 2-25 1-37
Comp-Att-Int 26-37-1 21-35-2
Sacked-Yards Lost 2-21 2-11
Punts 5-42.6 6-42.0
Fumbles-Lost 2-0 1-0
Penalties-yards 6-45 5-30
Time of Possession 36:17 23:43
Rushing • Washington, Morris 27-87, Cousins 3-22, Young 3-10, Royster 2-3. Cleveland, Richardson 11-28, Cribbs 1-16, Weeden 1-9, Hardesty 2-5.
Passing • Washington, Cousins 26-37-1-329. Cleveland, Weeden 21-35-2-244.
Receiving • Washington, Garcon 6-65, Moss 5-57, Morgan 4-60, Paulsen 4-47, Hankerson 2-56, Royster 2-30, Morris 2-6, Young 1-8. Cleveland, Little 5-74, Cameron 3-34, Gordon 3-27, Richardson 3-4, Massaquoi 2-16, Watson 2-12, Smith 2-8, Benjamin 1-69.
Missed Field Goals • None.