Matthew Stafford couldn't put the Lions ahead on either of their last two drives in a game he'd like to forget.
Stafford threw two interceptions one of which was returned 72 yards by Green Bay safety M.D. Jennings for a go-ahead score in the third quarter and lost a fumble.
Green Bay (7-3) moved a half-game behind first-place Chicago in the NFC North. The Bears play at San Francisco on Monday night.
Detroit (4-6) has dropped two in a row, making its goal of playing in consecutive postseasons very difficult.
Rodgers was 19 of 27 for 236 yards with two TDs, the first of which was a 20-yard connection with wide open Jermichael Finley early in the second quarter to give Green Bay its first lead, and an interception.
Cobb had nine catches for 74 yards and the game-winning TD, a twisting grab in the end zone. Finley had three receptions for 66 yards and a score.
James Starks helped the Packers strike somewhat of an offensive balance with 74 yards rushing on 25 carries.
Stafford completed just 17 of 39 passes missing many receivers badly for 266 yards with a 25-yard TD pass to Calvin Johnson that put Detroit ahead 17-14 late in the third quarter. He lost a fumble at midfield in the third quarter one drive after throwing a pass directly toward rookie cornerback Casey Hayward, who started for the injured Sam Shields.
In Detroit's last home game three weeks ago against Seattle, Stafford threw a winning TD pass with less than a minute left for the second time this season and for the fourth time in his four seasons.
He didn't come close to pulling off another rally.
The Lions lost 5 yards after Rodgers' second TD pass and on the final play of their next possession without a timeout, Stafford threw a 30-yard pass to Johnson and the receiver's lateral was recovered by safety Jerron McMillian at the Green Bay 30.
Johnson, who dropped some passes, had five receptions for 143 yards and caught a TD pass from Stafford for the second straight week after they failed to connect for a score in the first eight games of the season.
Mikel Leshoure had 84 yards rushing and a 1-yard TD run that gave Detroit a 10-7 lead midway through the second quarter.
The Lions settled for field goals after driving inside the Green Bay 10 early and late in the game and those missed opportunities proved to be costly.
If Green Bay lost, Crosby would've gotten a lot of the blame for staying in a funk in which he missed seven of 12 field goals. He missed a 50-yard field goal at the end of the first half failing to take advantage of a second chance after Detroit called a timeout to rattle him and missed a 38-yard kick midway through the fourth quarter.
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Green Bay 24, Detroit 20
Green Bay 0 7 7 10 24
Detroit 3 7 7 3 20
Det • FG Hanson 30, 4:24.
GB • Finley 20 pass from Rodgers (Crosby kick), 13:56.
Det • Leshoure 1 run (Hanson kick), 8:06.
GB • M.Jennings 72 interception return (Crosby kick), 8:31.
Det • Johnson 25 pass from Stafford (Hanson kick), 3:01.
Det • FG Hanson 27, 4:25.
GB • Cobb 22 pass from Rodgers (Crosby kick), 1:55.
GB • FG Crosby 39, :19.
Attendance • 63,716.
First downs 16 19
Total Net Yards 314 362
Rushes-yards 29-95 24-110
Passing 219 252
Punt Returns 0-0 1-0
Kickoff Returns 4-71 4-64
Interceptions Ret. 2-92 1-10
Comp-Att-Int 19-27-1 17-39-2
Sacked-Yards Lost 3-17 5-14
Punts 4-44.0 3-39.3
Fumbles-Lost 2-0 3-2
Penalties-yards 9-80 7-50
Time of Possession 30:18 29:42
Rushing • Green Bay, Starks 25-74, Cobb 2-19, Rodgers 1-3, Kuhn 1-(minus 1). Detroit, Leshoure 19-84, Stafford 2-12, Thomas 1-6, Bell 1-4, Smith 1-4.
Passing • Green Bay, Rodgers 19-27-1-236. Detroit, Stafford 17-39-2-266.
Receiving • Green Bay, Cobb 9-74, Finley 3-66, Nelson 3-45, Ja.Jones 2-33, Driver 1-12, Starks 1-6. Detroit, Johnson 5-143, Pettigrew 4-22, Broyles 2-35, Scheffler 2-22, T.Young 1-24, Heller 1-17, Bell 1-5, Leshoure 1-(minus 2).
Missed Field Goals • Green Bay, Crosby 50 (WL), 38 (WL).