Rodgers' performance gave a key victory to Green Bay (8-4), which remains one game behind Chicago for the top spot in the NFC North.
Six of Rodgers' completions went to Greg Jennings, who caught a 57-yard TD pass en route to 122 yards and two scores.
Donald Driver contributed four catches for 73 yards, including a memorable 61-yard scoring grab during which he eluded several would-be tacklers.
The 49ers (4-8) got a poor day from Troy Smith, who completed only 10-of-25 passes for 194 yards with a 66-yard touchdown pass to Vernon Davis and an interception.
2014: The Year in Music [PHOTOS]