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.
Ohio bar shooting arrested, charged with murder
LGBT community has 'bullied the American people': Bachmann