Rodgers was sidelined with a left shoulder injury suffered on the game's opening drive, forcing reserve Seneca Wallace into the lineup just as the Bears were left with McCown because Jay Cutler is out with a torn groin muscle.
Chicago's Matt Forte ran for 125 yards and a touchdown on 24 carries to spark the Bears to their first victory on the home field of their arch-rivals since 2007.
The result left the Bears, Packers and Detroit Lions all level at 5-3 atop the NFC North division, with the Lions visiting Chicago next week while Green Bay hosts Philadelphia.
Rodgers suffered the injury to his non-throwing shoulder in the first quarter when he was sacked by Chicago's Shea McClellin.
The Packers suffered their first home loss since last year's season opener against San Francisco despite 150 rushing yards and a touchdown from Eddie Lacy and 114 yards on 11-of-19 passing for Wallace, who had seldom played in the past two years.
Alshon Jeffery caught a 6-yard touchdown pass from McCown with 2:41 remaining in the third quarter to give the Bears the lead for good.
Chicago sealed the victory on an 18-play, 80-yard drive that took almost nine minutes and ended with Robbie Gould kicking a 27-yard field goal with 50 seconds to play.
The Bears now lead the NFL's oldest rivalry 92-87 with six games drawn.