Smith hooked up with Dexter McCluster and Anthony Sherman on scoring strikes, completing 24-of-36 passes for 225 yards in the victory.
Jamaal Charles provided 120 yards of total offense for the Chiefs (8-0), who built a 20-7 lead but then had to hang on down the stretch to continue their best start since 2003.
The Browns (3-5) dropped a third straight despite a strong performance from Jason Campbell, who amassed 293 passing yards with two touchdowns on 22-of-36 efficiency.
Josh Gordon led Cleveland's receiving corps, making five grabs for 132 yards and a score.
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