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However, the Cowboys quickly responded in kind when Dan Bailey's kick from 51 yards flew between the posts.
Brady tried to restore New England's lead, but the Dallas defence continually targeted the Pats quarterback, sacking him four time before the halfway point in the second quarter.
Indeed, one of those sacks forced a fumble from the future Hall of Fame star, but Julian Edelman was on hand to recover the football before the Patriots were forced to punt later on the drive.
Brady bounced back with a quarterback sneak to give the visitors a 10-3 lead with the break approaching.
However, New England extended that advantage by three after Gostkowski produced a career-long field goal of 57 yards, which cleared the crossbar with plenty to spare. The kick also matches Adam Vinatieri's mark for the longest in Pats history.