Richardson scored on a pair of 1-yard runs en route to 42 yards on 18 carries while Weeden went 17-of-30 through the air as the Browns (5-8) picked up their third straight win.
Former Brown Brady Quinn could manage only 11-of-22 passing for 159 yards and a interception against his former team for the Chiefs (2-11), who lost for the seventh time in eight games.
Jamaal Charles amassed 165 yards with a score on 18 carries to provide Kansas City's lone standout performance.
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