X-rays at Scott Stadium confirmed the fracture.
"Our kids are happy and they're celebrating the win, but they lost a kid they feel really strongly about," Kelly said after the game. "The coaches are probably feeling it more. I am in particular because I'm so close to him. I feel bad for him."
Kelly said Sunday that Zaire, who will begin rehab once swelling goes down, should be back for the Irish in the spring.
The redshirt sophomore quarterback was making his third career start. He had completed 7 of 18 passes Saturday for 115 yards and a touchdown, while rushing 10 times for 87 yards.
Redshirt freshman DeShone Kizer will take over as the team's starting quarterback, Kelly said. On Kizer's first play after Zaire's injury, he handed off to running back C.J. Prosise for a 24-yard touchdown run.
Kizer finished with 92 yards and two touchdowns on 8-of-12 passing, including a 39-yard go-ahead strike to wide receiver Will Fuller to silence the Cavaliers' comeback bid.
"He can win games for us," Kelly said of Kizer. "I want to win a championship with him. We need to get him to the next level."
Kelly said Sunday that Malik will continue to be around the team and will travel with the Irish for road games.
"We've lost our starting RB and starting QB, but there were a lot of good players around them. DeShone isn't going to have to win it himself," Kelly said.
ESPN.com's David Hale and Matt Fortuna contributed to this report.