He will have surgery and is out indefinitely.
Cooley was injured when he took a big hit from Eagles cornerback Nnamdi Asomugha in the first quarter.
Cooley went to the locker room after the hit, but returned to the game briefly before leaving the game again. He was then seen on the sidelines without his jersey and shoulder pads.
The Redskins, who switched quarterbacks at the start of the fourth quarter, were also dealt serious injuries on the left side of their offensive line.
Cooley has been bothered by a left knee injury most of the season. He was questionable for Sunday's game against the Eagles after having his knee drained again during the week. He has only seven catches for 66 yards.
Fourth-year player Fred Davis has taken the majority of Washington's snaps at tight end in Cooley's absence. Davis led the Redskins in receiving Sunday, hauling in nine passes for 95 yards in the loss.
Information from The Associated Press contributed to this report.