Rodgers' 1-yard scoring strike to Lee with 12:42 left in the game broke a 3-3 tie and capped a five-play, 75-yard drive highlighted by a 46-yard bomb to Greg Jennings to the Bears' 1-yard line.
The Packers then used a tough defense to stymie Chicago the rest of the way and sealed the win when Nick Collins intercepted a pass with 10 seconds remaining, sending the Packers into the NFC playoffs with the No. 6 seed.
They'll play the NFC East champions Philadelphia Eagles next week.
Rodgers ended with 229 yards passing and Jennings hauled down four passes for 97 yards for the victors.
The Bears (11-5) played their starters despite having nothing to play for in post-season seeding. They retained their second seed in the NFC despite the loss.
Jay Cutler passed for 168 yards with two interceptions and Matt Forte gained 91 yards on 15 carries for Chicago.