Costa left on a cart with an air cast on his leg in the second quarter. X-rays were negative, but Costa was in a walking boot after the game and will undergo more tests Monday.
Cowboys linebacker Sean Lee also left the game with a sprained toe and will have an MRI on Monday.
"I'm not a doctor but I think it should be good," Lee said.
Costa injured his ankle while attempting to recover after Carolina returned Miles Austin's lost fumble. Costa immediately grabbed his ankle and rolled on the ground.
"It looked very severe to me from where I was sitting, and when they bring the cart out and they immobilize it all of those things," Cowboys coach Jason Garrett said. "He's a tough kid. When he came inside, it still looks like a pretty significant injury. We'll monitor it day-to-day, but it looked really severe when it first happened."
Ryan Cook took over for Costa and would assume the starter's job in his absence, like he did for three games.
Costa only returned to the lineup last week against Baltimore after missing three games with a back strain. He was slowed by a knee injury earlier in the week during practice. Costa's season started slowly because of a back strain suffered in training camp, and he missed three preseason games.
Cook, whom the Cowboys traded for five days before the season started, took over for Costa in the season opener after one series when Costa aggravated the back injury. Cook started the next three games but suffered a strained hamstring on Oct. 1 against Chicago.
Lee had his foot re-taped, but he was not effective while attempting to do some running on the sideline and had his foot taped again. Lee was limping after the game and wore tennis shoes with his suit as he walked to the bus.
Cowboys wide receiver Dez Bryant missed one offensive series after getting hit on the head during a 10-yard punt return in the third quarter.
Bryant did not have a catch the rest of the game and dropped a touchdown pass in the fourth quarter that would have given the Cowboys a 20-14 lead. Bryant said he was fine after the game.