Under first-year coach John Fox, the Panthers (6-9) got off to a 3-0 start before enduring an eight-game losing streak. But they have won three of four since that skid.
"These guys have worked hard. I have been proud of their efforts," Fox said. "I think we've made strides. We are trying to build something. We want to finish strong."
An undrafted rookie out of Syracuse, Brown helped the Panthers to their latest win, scoring on runs of two and one yard in a 21-point second quarter.
Peete added a three-yard touchdown pass to Muhsin Muhammad in the second-quarter uprising.
The Bears (4-11) were forced to use untested Henry Burris at quarterback for much of the contest after veteran Chris Chandler suffered a concussion in the second quarter.
Burris completed just eight of 22 passes for 50 yards and a touchdown. Chandler was seven of eight for 113 yards before he was injured, including a 76-yard touchdown pass to Dez White.
Looking for their second straight win, the Bears got off to a flying start when Chandler found White for his 76-yard touhdown pass on the game's second play.
But it was all downhill from there as the Bears watched Chandler wooble to the sideline with under 11 minutes left in the first half when took a late hit from Brenston Buckner, who was penalized for roughing.
Once Chandler departed, so did the Bears' passing game. Burris failed to generate much of anything and also lost two fumbles.
"That's a situation you have to understand your job and responsibility," said Burris about replacing Chandler. "You have to be ready for that type of situation. Things started to get a little more comforting as the game went on. I was able locate things a lot quicker and if nothing was there, I tried to move around to make things happen."
Chicago also could not run the ball as Leon Johnson was held to 29 yards on 16 attempts.
Peete helped the Panthers get even in the second quarter, connecting with Steve Smith on a 69-yard pass to the Chicago two. Brown ran for a touchdown on the next play.
The second of the Bears' four fumbles helped set up the go-ahead score when Ahmad Merritt muffed a punt and Emanuel McDaniel recovered for the Panthers at the Chicago 11.
Three plays later, Brown ran one yard for a touchdown, giving the Panthers a 14-7 lead with 8:38 remaining in the first half.
Another turnover helped the Panthers extend the lead when Burris was sacked by Al Wallace and rookie Will Witherspoon recovered for Carolina at the Chicago 41.
"The game will never change," Bears coach Dick Jauron said. "If you turn the ball over as many times as we did, particularly in the field position we turned it over, you're not going to win the game. That's what happened today."
Brown had runs of 12 and two yards and a 22-yard interference penalty on Roosevelt Williams move the ball to the five. Two plays later, Peete found Muhammad for a three-yard score, extending the lead to 21-7 with 3:52 left before intermission.
Peete finished 18 of 35 for 179 yards.
It was a mistake by Peete that helped set up the Bears' second touchdown early in the fourth quarter.
On third and seven from his own 44, Peete was hit by linebacker Bryan Knight and fumbled. Star linebacker Brian Urlacher picked up the loose ball and returned it 43 yards to the Carolina eight.
Burris capped the short drive with a five-yard touchdown pass to White with 11:48 remaining, pulling the Bears within 21-14.
But Burris' second fumble deep his own territory cost the Bears any shot at a comeback. At his own 20, Burris was sacked by Mark Fields, who recovered at the seven.
That miscue led to Shayne Graham's 25-yard field goal, giving Carolina a 24-14 lead with 5:16 to play.