Debuchy limped off after just 16 minutes of his first appearance for Arsenal since November 2015 and Giroud sustained a hamstring problem.
Gunners boss Wenger revealed the injury-plagued Debuchy fears he has a serious injury and the right-back is set to go for scans next week to determine the extent of the problem.
"Debuchy thinks it is a severe one, but don't go too far yet. I have to speak to the medical people," Wenger said.
"We have to let it breathe. Usually he would have an MRI in a couple of days.
"Debuchy had many setbacks so I imagined something could happen to him when I picked the squad."
Giroud will also be assessed to see how bad the damage is following the striker's substitute appearance, but Wenger played down concerns over Theo Walcott, who came off following a strong second-half tackle.
"Walcott was just a kick. The only worry we have is Giroud," Wenger said.
"He looked to touch his hamstring. We have to see how deep the damage is. He told me on his first movement he felt a twinge."