Dalvin Cook gained 130 yards from scrimmage and scored two touchdowns.
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Cousins finished with 242 yards and one TD passing, highlighted by his 43-yard completion to Adam Thielen on the opening possession of overtime to give Minnesota the ball on the Saints 2-yard line and set up the winning score.
Saints quarterback Drew Brees passed for 208 yards and a touchdown, but had two turnovers.
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Vikings are set to play San Francisco 49ers in NFC Divisional game on Saturday, Jan. 11, at Levi Stadium in Santa Clara.