Claiborne suffered the noncontact injury on a third-and-3 play with 5:52 left in the first quarter. He did not return to the game, and owner and general manager Jerry Jones said initial fears were a torn anterior cruciate ligament.
Cowboys coach Jason Garrett said Claiborne will have surgery in a few days.
"Obviously he's disappointed," Garrett said. "Mo Claiborne is going to be a good player in this league. There's no doubt in my mind. Anybody that's been around him I think believes that and he's working hard to get there. Injuries have been a part of his progression here so far. I have been around players in my career that have had similar starts to their careers and gotten themselves healthy and gotten themselves stronger and ready to play at this level. They've had really good careers after that.
"Mo has been dealing with a lot of different things from a physical standpoint. This is another one of those. He's going to get surgery done and he's going to come back stronger and better than ever."
The Cowboys have used their short-term injured reserve designation on rookie defensive end DeMarcus Lawrence, who had foot surgery in training camp.
The injury capped a tumultuous week for the cornerback. Despite a game-sealing interception against the St. Louis Rams, Claiborne was demoted after giving up two touchdowns, including a 51-yarder, and a 38-yard catch.
Claiborne walked out of the team's Valley Ranch facility on Tuesday after he was informed of the team's decision to start Orlando Scandrick but returned that night to meet with Garrett. Claiborne apologized to the team the next day and actually started Sunday's game against the Saints with the Cowboys opening up in their nickel defense.
He played in four of the first six snaps before getting hurt. Sterling Moore replaced Claiborne and finished with three tackles, two pass deflections and a fumble recovery.
The Cowboys have only four cornerbacks on the 53-man roster with Brandon Carr, Scandrick, Moore and undrafted rookie Tyler Patmon. They could look to call up Micah Pellerin from the practice squad or sign a veteran if they want another corner.
The Cowboys had moved up to the sixth overall pick of the 2012 draft to select Claiborne, who has one year left on his rookie contract that guaranteed him $16.26 million. His $2.607 million base salary in 2015 is fully guaranteed. The Cowboys have until next spring to pick up the fifth-year option on his contract for 2016, which they almost assuredly will not do.
In 29 games, Claiborne has recorded three interceptions and 18 pass deflections.