The Packers got the two touchdowns on their first two possessions. Rodgers directed a seven-play 78-yard opening drive.
Green Bay's second drive went for 68 yards on 11 plays, capped by a two-yard run by Grant.
Rodgers threw for 102 yards on 5-of-10 passing. Backup QB Matt Flynn hit on 5-of-6 passes for 50 yards.
The Browns were held to 11 first downs and 191 total yards and suffered three sacks.
Brady Quinn, the 2007 first-round draft choice, started the game and was 7-for-11 passing with one interception and 68 yards.