The linebacker from Connecticut, the 63rd overall pick, was injured in the Cardinals' 17-6 preseason loss to the San Diego Chargers on Saturday night.
Coach Ken Whisenhunt said the wrist was put back into place after the game but will require surgery to repair ligament damage once the swelling goes down.
There was some good news, though, on Monday when Arizona's first-round draft pick, running back Beanie Wells, had his best workout since spraining his right ankle in his first training camp practice.
"He looked good today. He looked explosive, looked fast, moved around well out there on the field," Whisenhunt said. "Hopefully that will progress during the week and we can get some good work out of him in the game this week."
"There were a couple of runs where he got on the perimeter and really hit it," Whisenhunt said of Wells at practice, "and that's what you want to see because it's really a process of getting comfortable with where your own ankle is and being able to make those cuts."
Brown, who started all 26 games as a junior and senior at Connecticut, played defensive end in college and was making the switch to outside linebacker with the Cardinals.
"Where we were looking to see the biggest contributions from him was on special teams," Whisenhunt said. "He was making some good strides there. ... and certainly we had expectations of using him as a situational pass rusher as well."
Starting cornerback Bryant McFadden, who missed Saturday's game with a sore ankle, practiced some on Monday.
"He's still a little bit sore so it was more precautionary with him," Whisenhunt said.
Steve Breaston, who bruised his right knee in the preseason opener at Pittsburgh, is close to returning to practice, Whisenhunt said.
Quarterback Brian St. Pierre sat out practice with a sore back but only as a precaution, the coach said.
Meanwhile, quarterback Kurt Warner said his surgically repaired hip feels much better than it did when training camp began.
"It probably feels twice as good as I've felt at any point since I injured my hip," he said. "It's exciting for me that I'm seeing strides being made, I'm feeling better, I'm feeling comfortable, I don't have to worry about trying to manage the pain."
The Cardinals worked out in the desert heat at their practice facility. They had to end their training camp in much cooler Flagstaff a week earlier than usual because classes began at Northern Arizona University, where the team stayed.