Rodgers threw for 312 yards and three touchdowns. Brees threw for 419 yards and three TDs, including a late touchdown to Jimmy Graham that cut the lead to eight with 2:15 left.
After a Green Bay punt, Brees marched the Saints to the Packers' 9 and spiked the ball with 3 seconds left. Green Bay's A.J. Hawk was called for pass interference and the ball was placed at the 1.
The Packers then swarmed rookie running back Mark Ingram, and the game was over.
It was a big night for the Packers' Donald Driver, who tied James Lofton's franchise mark for career yards receiving with 9,656. He caught four passes for 41 yards.
Rodgers threw three first-quarter TD passes - finding Greg Jennings, Jordy Nelson and Cobb - as Green Bay's offense picked up where it left off after a 31-25 win over Pittsburgh in the Super Bowl.
Before the game it was reported that no progress had been made on a contract extension for Brees. The team had hoped to get a deal done before the season started. But ESPN, citing a source familiar with the talks, reported the sides were far apart.
Brees is in the final year of a 6-year, $60 million deal.