The Packers joined the New Orleans Saints, San Diego Chargers and Philadelphia Eagles in claiming playoff spots on the final day of the regular season.
Philadelphia beat Dallas 24-22 in a head-to-head battle for the NFC East division crown and a playoff spot, while New Orleans secured a post-season position with a 42-17 rout of Tampa Bay, and San Diego edged Kansas City 27-24 in overtime to seal a playoff berth.
Rodgers, who missed the past seven games with a broken collarbone suffered in a loss to Chicago last month, connected with wide-open Cobb, who had been out for more than two months with a knee injury, to convert a third consecutive fourth-down play on the winning drive.
"This is an amazing feeling. What a game," Rodgers said. "So many plays. I felt like I was in a rhythm, like we were moving good."
The Bears saw their season end on an interception off a desperation pass on the game's final play as the Packers took the NFC North division title.
"You have to give the Packers credit," Bears coach Marc Trestman said. "They were able to convert three fourth downs and Aaron Rodgers was able to make a great play to win the game."
In the American Conference, Denver captured home-field advantage throughout the playoffs with a 34-14 romp at Oakland.
New England joined Denver with a first-round bye after defeating Buffalo 34-20.
In the National Conference, Seattle beat St. Louis 27-9 to secure a home-field playoff edge and first-round bye, while NFC South champion Carolina clinched a bye as well with a 21-20 win at Atlanta.
Four teams chased the final American Conference berth with Pittsburgh beating Cleveland 20-7 to force San Diego to win or miss the playoffs.
The host Steelers improved to 8-8, avoiding their first losing season since 2003, but needed losses by Miami, Baltimore and San Diego to reach the playoffs.
Reigning Super Bowl champion Baltimore was eliminated with a 34-17 loss at Cincinnati, the Ravens missing the playoffs for the first time since 2007.
Miami lost 20-7 at home to the New York Jets, leaving both teams 8-8 and out of the playoffs on tie-breakers.
That left the Chargers needing only a draw against the Chiefs, who benched most of their starters having already clinched a playoff berth.
San Diego trailed by 10 points in the fourth quarter before pulling level at 24-24 on Nick Novak's 22-yard field goal with 3:21 to play and winning on Bovak's 36-yard field goal in overtime.
Tennessee beat Houston 16-10, ensuring the defeated Texans the top pick in next year's NFL Draft after a league-worst 2-14 season.