The Frenchman, who joined Arsenal in July for a fee believed to be around £12 million, suffered the injury during the Gunners’ 2-2 draw with Manchester City earlier in the month.
It was initially hoped that Debuchy had only suffered a sprain, but further tests revealed ligament damage and he will now spend a substantial period out of action.
“He had surgery on Friday and will be out for three months,” said manager Arsene Wenger.
“A reasonable delay is three months. All went well, but it can be a week or two earlier or a week or two later, but it will be around three months.
“It is a blow of course, how big it will be we will only know afterwards because it will be down to how well we replace him. We bought [Calum] Chambers for that.”