Cousins then threw a touchdown pass to Kyle Rudolph for the win.
Minnesota advanced to the divisional round of the playoffs for the first time since the 2017 season. That season, the Vikings lost in the conference title game to the Philadelphia Eagles.
Cousins finished the game 19-for-31 with 242 passing yards and the game-winning touchdown passes to Rudolph. Running back Dalvin Cook had 28 carries for 94 yards and two touchdowns.
Minnesota had a 10-point lead heading into the fourth quarter but the Saints battled back with a Taysom Hill touchdown catch from Drew Brees with 10:31 remaining in the fourth.
Saints kicker Will Lutz then added the game-tying 49-yard field goal.
Brees finished the game 26-for-33 with 208 passing yards, a touchdown pass, an interception and a key lost fumble late in the game.
Saints wide receiver Michael Thomas had seven catches for 70 yards. Tight end Jared Cook had five receptions for 54 yards.
The Vikings will travel to play the San Francisco 49ers next week.
The Saints fell short of the Super Bowl for the third consecutive season.