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Nelson was involved again on the next drive, moving the Packers to within a couple of yards of the endzone before Eddie Lacy carried it in on the next play.
The Packers continued to run the ball, and added a third first-quarter touchdown through a James Starks run as the Panthers continued to struggle.
Cobb scored his eighth touchdown of the season shortly before half time as Green Bay piled more misery on Carolina, but Graham Gano was able get the Panthers off the mark with a field goal just before half time.
The Packers' dominance continued in the second half as Davante Adams made a diving catch from a Rodgers pass for the fifth touchdown of the day for the home side.
The Panthers had to play the majority of the second half without last season's defensive player of the year after Luke Kuechly was ejected for lashing out at an official.
Carolina did add a couple of consolation touchdowns late on after the Packers chose to rest Rodgers, with Kelvin Benjamin collecting a pass from Newton before Brenton Bersin linked up with backup quarterback Derek Anderson.
The result moves the Packers onto 5-2, while the Panthers slip to 3-3-1, but remain at the top of the NFC South.