The Packers were without star receivers Randall Cobb and James Jones, and tight end Jermichael Finley was carried off the field on a stretcher in the third quarter after taking a hard shot to the head from Cleveland safety Tashuan Gipson.
Rodgers completed eight passes for 103 yards and a score to second-year wide receiver Jarrett Boykin. Overall, he threw for 260 yards on 25-of-36 efficiency for the Packers (4-2), who ran off their 10th consecutive home win.
Finley hauled in a scoring strike before exiting and Jordy Nelson caught a touchdown pass for Green Bay.
Cleveland (3-4) dropped a second straight as Brandon Weeden was limited to just 17-of-42 through the air for 149 yards with a TD and an interception.
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