"The decision was not my own...," Griffin tweeted. "But I will be there for my team!!! In every way I possibly can."
Redskins players were informed of the decision at a team meeting Saturday night, someone in the meeting room texted to ESPN minutes after players were told.
Asked earlier in the day if he had a gut feeling whether Griffin (Grade 1 sprain of the lateral collateral ligament) would be available, coach Mike Shanahan responded, "No, not really. We'll get a chance to evaluate him (on Saturday) and talk to the doctors."
Apparently, doctors weren't entirely comfortable, as much as Griffin wanted to play.
The Redskins did not say whether Griffin will be active for the game, or whether veteran Rex Grossman, who has been an inactive third-string quarterback for every game this season, will serve as the primary backup to Cousins, who will be making his first career start.
Shanahan had been coy with his game plan heading into Sunday, not wanting to rush Griffin's recovery or surrender a competitive advantage to the Browns.
Griffin practiced in a limited capacity Wednesday through Friday and is listed as questionable for Sunday. While Shanahan resisted giving percentages of practice repetitions, he did say Griffin looked better each day.
Griffin, who is tied with New England's Tom Brady for the NFL lead in passer rating and whose popularity has soared in a few short months as a 22-year-old newcomer to the league, was injured when he was hit by defensive tackle Haloti Ngata late in regulation during last week's 31-28 overtime victory over the Baltimore Ravens.
An MRI revealed the mild sprain of the LCL.
Cousins was drafted in the fourth round out of Michigan State, a surprise selection given the Redskins already had selected Griffin with the No. 2 overall pick. He made his first appearance after Griffin suffered a concussion against the Atlanta Falcons in Week 5 and threw one touchdown pass and two fourth-quarter interceptions in a 24-17 loss.
Then last week, Cousins went 2 for 2 in the Redskins' final drive of regulation after Griffin was injured, throwing an 11-yard touchdown pass to Pierre Garcon with 29 seconds left and then running in the game-tying 2-point conversion.
Cousins has completed 7-of-11 passes for 137 yards with two touchdowns and two interceptions in his two appearances. He has gained a reputation as a studious backup, benefiting from the fact that he and Griffin were brought along together in an offseason program and training camp specifically designed by Redskins coaches for quarterbacks new to the pros.
The Redskins were 3-6 before their bye week, but now they are one game behind the Giants and also one game behind the front-runners for the NFC's two wild-card berths. Griffin has led the comeback, throwing for 10 touchdowns and only one interception during the four-game winning streak.
Information from ESPN NFL Insider Adam Schefter and The Associated Press was used in this report.