Flacco, whose wife gave birth to the couple's second child just hours before the game, helped the Ravens (1-1) notch their first win of the season and remain unbeaten in 11 all-time meetings with the Browns.
The reigning NFL champions played their first home game since defeating San Francisco in Super Bowl XLVII, which was marked by pre-game ceremonies.
Bernard Pierce and Marlon Brown each scored a second-half touchdown to secure the victory.
The Browns (0-2) lost quarterback Brandon Weeden to an injured thumb after he had thrown for 227 yards.
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