Cousins threw for 329 yards and two touchdowns, leading the Redskins to a 38-21 victory at Cleveland for their fifth straight victory.
Grossman has been inactive for every other game this season. After the game, Griffin said he felt he could have played if the Redskins had needed him.
"I was not happy with the decision," Griffin said. "That's the decision they went with and I respect that. It doesn't mean I have to necessarily like it. I feel like I could have played this week, next week, the week after. But that's not my decision."
Redskins players were informed of the decision to start Cousins at a team meeting Saturday night, someone in the meeting room texted to ESPN minutes after players were told.
Coach Mike 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.
Information from ESPN NFL Insider Adam Schefter The Associated Press was used in this report.