On McCoy's second play from scrimmage - and his first pass attempt - he threw a short pass to Pierre Garcon that the receiver turned into a 70-yard touchdown.
Washington took a 13-10 lead with the play.
It was the longest TD pass of McCoy's career and his first scoring toss in the NFL since December 2012 for Cleveland.
Cousins had a fumble and interception in the first half, raising his giveaway total to 11 this season. He was 10 of 16 for 139 yards in the half.
Cousins has been starting for Washington while Robert Griffin III is sidelined with a dislocated left ankle.
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