McCown, subbing for the injured Jay Cutler, completed 22-of-41 passes for 272 yards, including a go-ahead, 6-yard scoring strike to Alshon Jeffery late in the third quarter.
Matt Forte amassed 125 rushing yards with a score on 24 carries and also reeled in five passes for 54 yards as the Bears posted a victory in Green Bay for the first time since 2007.
Brandon Marshall made seven grabs for 107 yards and a touchdown for Chicago, which snapped Green Bay's 10-game home winning streak.
With the win, the Bears, Packers and Detroit Lions knotted for the NFC North lead with identical 5-3 records.
Rodgers left during the game's opening possession when he was sacked by Shea McClellin and injured his non-throwing shoulder.
Seneca Wallace replaced him behind center and went 11-of-19 through the air for 114 yards with an interception.
Eddie Lacy amassed 150 rushing yards with a TD on 22 carries.