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Mark Sanchez started the game for the Eagles in place of the injured Sam Bradford, and he was able to lead his side to a touchdown on their first possession when he threw for Josh Huff, who made a cut to beat two defenders and race in for a 35-yard score.
The Buccaneers rallied after Doug Martin raced through the line to rush 51 yards downfield to set up a touchdown grab for Mike Evans as he collected Winston's pass by tapping his toes down in the corner of the endzone.
Vincent Jackson handed the visitors the lead in the contest when Winston drilled a 13-yard strike to him for a score as he managed to beat man coverage on the goalline, while an 84-yard gain from Martin allowed Winston to notch his third touchdown of the game by finding Russell Shepard.
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The Eagles rallied as Sanchez was able to find Darren Sproles for a short pass, and the running back made three players miss with attempted tackles to race into the endzone for a 35-yard score.
However, Tampa Bay restored a 14-point advantage before the break when Winston looped the ball to the right side of the endzone for Charles Sims to beat two players in coverage for the touchdown.
The Buccaneers piled on the misery for the Eagles as Winston threw for his fifth touchdown of the match when he picked out tight end Cameron Brate for an eight-yard score to extend the visitors' lead to 21 points.
Connor Barth ensured that his side would secure the victory when he made his first field goal of the game by sending his kick between the posts from 28 yards, while Caleb Sturgis added a consolation kick for the Eagles from 43 yards.
Tampa Bay capped their win when Sanchez was intercepted for the third time of the match as Lavonte David made the pick and returned the ball 20 yards for a touchdown.