Rodgers, playing in his first game since breaking his collarbone Nov. 4, was facing a fourth-and-8 situation with 38 seconds left when he eluded Chicago's Julius Peppers and spotted a wide-open Cobb for the game-winning score.
Rodgers finished 25-for-39 for 318 passing yards, two touchdowns and two interceptions as the Packers (8-7-1) claimed the NFC North title with a third win in four games to finish the regular season.
Cobb, himself returning from a fractured right tibia, made two scoring grabs in all.
Jordy Nelson led the Packers' receiving corps with 10 catches for 161 yards.
Jay Cutler made good on 15-of-24 passes for 226 yards, two touchdowns and an interception for Chicago (8-8), which ended with two straight losses to miss the playoffs for a third consecutive season.
Matt Forte scored three times and piled up 157 total yards for the Bears.