Weeden, one of Cleveland's two first-round draft choices, started the game and played for three drives, gaining 62 yards while tossing one interception.
The Browns' other 2012 first-round draft pick, running back Trent Richardson, missed the game while rehabbing from minor knee surgery.
Adonis Thomas rushed for touchdown and Seneca Wallace passed for another in Cleveland's victory.
Browns wide receiver Mohamed Massaquoi left the game with a head injury.
2014: The Year in Music [PHOTOS]