Kenyon Coleman has torn triceps
An MRI on Monday confirmed the injury, and the 33-year-old veteran will have the surgery Tuesday.
"He'll be a loss for us," Garrett said. "But like we've had with other guys who've gone out this year, the next man's got to be up. We feel like we've got some good depth on the defensive line position, so the next guy is going to jump in there."
Coleman suffered the injury during LeSean McCoy's 23-yard run on Philadelphia's first drive of the second half.
Garrett would not indicate who would start for Coleman, but the candidates are Tyrone Crawford, Marcus Spears and Sean Lissemore, who could come back from a high left ankle sprain after missing most of the last five games.
Garrett said the team would look to the practice squad for a replacement in either Ben Bass or Robert Callaway.
The injury could put Coleman's career with the Cowboys in question. He will turn 34 in April and is set to be an unrestricted free agent. He started seven games this season, missing the Chicago and Baltimore games with a knee injury.
Church suffered a torn Achilles tendon on Sept. 23 against Tampa Bay, and Lee suffered a torn plantar plate in his right big toe on Oct. 21 at Carolina. Both players are rehabbing from surgeries.