Costa, a fifth-year senior and leader of the top-ranked Ducks, was hurt when he bobbled a field-goal attempt -- he also serves as the holder -- and was tackled as he tried to scramble to the sideline with the ball. He was taken from the field on a cart.
The injury effectively ends Costa's college career. He said he knew it even before he had an MRI on Monday.
"To know my college career has come to an end because of this injury made it a little harder," he said Tuesday. "But I've got a lot of stuff ahead of me in the future and it's pretty bright so I'm looking forward to that."
Costa threw for 151 yards and a score, and ran for another TD, against Washington State earlier this season when Darron Thomas left with a shoulder injury.
Costa finishes this season with 286 yards passing and a touchdown. He was also Oregon's primary holder for the past two seasons.
"He talked to our team this morning and told them not to feel sorry for him," Oregon coach Chip Kelly said. "He'll still be with us the rest of the way."
Costa battled with Thomas during fall camp for Oregon's starting job after last season's quarterback, Jeremiah Masoli, was dismissed from the team in the offseason.
The loss of Costa poses a dilemma for the Ducks, who have a recent history of inopportune quarterback knee injuries.
Oregon was hoping to redshirt freshman Bryan Bennett, who was impressive during fall camp. But instead he was listed as Thomas' backup for Saturday's game at California.
The team also has redshirt freshman Daryle Hawkins, who serves as a reserve quarterback, receiver and running back.
Kelly said Costa's injury does not change Oregon's game planning. He also said he would not make a special effort to get Bennett playing time in a game.
But Kelly made it clear that he would not hesitate to call on the freshman.
"Our goal is to win the football game," Kelly said. "If we need to play Bryan Bennett in the football game then that's what we're going to do."
Costa will travel with the team to Strawberry Canyon.
"I'm going to maintain a leadership role with this team whether I'm playing or not," he said. "That's something I can help with."
The lack of an experienced backup for Thomas is a concern. Oregon saw a chance to compete for a national title crumble in 2007 when Dennis Dixon went down with a knee injury in the third-to-last game of the regular season.
That season, backup Brady Leaf was also knocked out with a knee injury.
Costa has sustained three serious knee injuries while at Oregon, two to his left and the latest to his right.
For his career, Costa completed 50-of-71 passes for 556 yards, two touchdowns and one interception. He also rushed for 195 yards and three scores.