Cousins, subbing for injured fellow rookie quarterback Robert Griffin III, found Leonard Hankerson on both of his touchdown passes as the Redskins (8-6) claimed a share of first place in the NFC East with a fifth straight win.
Washington pulled into a three-way tie with the New York Giants and Dallas Cowboys atop the division following the Giants' loss to Atlanta and the Cowboys' win over Pittsburgh.
Alfred Morris rushed for 87 yards and two touchdowns on 27 carries in the win.
Cleveland (5-9) saw its three-game win streak snapped as Trent Richardson was held to 28 rushing yards on 11 carries with a pair of touchdowns.
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