Cousins is in his second season in the NFL. He's appeared in two games this season, completing 12-of-25 passes for 107 yards and two interceptions. He has yet to throw a touchdown pass. He's had one previous NFL start, beating the Cleveland Browns 38-21 last December.
Cousins is listed as the starter and Rex Grossman as his backup for Sunday at Atlanta. Both teams are last in their divisions with 3-10 records. Only the Houston Texans (2-11) have a worse record this season.
Griffin is expected to be listed as inactive.
"We made a decision that I think is best for Robert," Shanahan said.
He said he wants the quarterback healthy for off-season preparation ahead of the 2014 season. He said the second and third year of off-season work is important for the development of an NFL quarterback.
Shanahan said he "didn't think it was worth the risk" for Griffin to continue playing. Griffin had off-season knee surgery that appeared to hamper his play early in the year but he played in all 13 games so far this season. Griffin is 274-of-456 passes for 3,203 yards, 16 touchdowns and 12 interceptions in 2013.
Griffin, who won the 2011 Heisman Trophy while playing for Baylor and was named to the Pro Bowl as an NFL rookie in 2012, also has been sacked 38 times. Shanahan said he was concerned about the cumulative effect of the hits Griffin has taken this season.
"I'm sure everyone knows by now coach decided to shut me down for the rest of the season," Griffin said Wednesday. "I expressed my desire to play. ... He explained to me his reasoning and at the end of the day, coach's decision is what we go with."
Cousins said: "Robert is the franchise quarterback. This is his team. My job is to help this team get a win."