Cousins completed a touchdown pass on his first snap of the day, picking out full back Darrel Young for a 20-yard completion.
Washington lost another of their offensive starters later in the first quarter as DeSean Jackson landed awkwardly on his shoulder.
Chad Henne could not get the Jaguars in a position to get on the board in the first quarter, allowing Cousins to get the ball back down the other end before Alfred Morris carried for a one-yard rushing touchdown at the start of the second.
The running back powered over for another later in the half, but Henne connected with tight end Marcedes Lewis just before the break to keep the Jaguars in contention.
The Redskins have confirmed that neither Griffin nor Jackson will return to the game.