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Brown put the Giants ahead in the first quarter with an impressive field goal from 41 yards out.
The Lions responded with a kick of their own in the second quarter when David Akers contributed from 37 yards to bring the score level at 3-3.
The Giants pushed themselves ahead thanks to a touchdown from Jerrel Jernigan, who was able to latch onto a 20-yarder from Eli Manning, before Brown kicked another field goal from 52 yards to leave the score at 13-3 going into the interval.
After the break, Detroit's Joique Bell produced a one-yard run for the touchdown, before the Lions gained an extra two points following a fumble, leaving the score at 13-12 going into the final quarter.
However, Will Hill completed a 38-yard interception return to force overtime, in which Brown stepped up again to kick for victory from 45 yards out.