The Vikings had other plans. They ran the ball often and well, led by running back Dalvin Cook, who had 94 yards rushing and two touchdowns. Their running game helped keep the Saints’ high-powered offense off the field and created opportunities for Cousins to find open receivers, most notably Thielen, who caught seven passes for 129 yards.

At the same time, the Vikings’ defense stymied the Saints’ biggest offensive weapons. Brees completed 26 passes for 208 yards and one touchdown. But he threw an interception in the second quarter that the Vikings later turned into a touchdown, and he fumbled the ball while being sacked with under five minutes left in the fourth quarter, allowing the Vikings to burn time off the clock.

Alvin Kamara, the Saints’ explosive running back, had just 21 yards rushing and 34 yards receiving. Saints receiver Michael Thomas, who caught an N.F.L.-record 149 passes this season, had seven receptions for just 70 yards. Wil Lutz, the team’s reliable kicker, missed a 43-yard field-goal attempt just before the first half ended.

“We just fell short on offense,” Saints tight end Jared Cook said. “We didn’t do our job.”

Still, heading into the fourth quarter behind, 20-10, the Saints found life. Brees completed passes to several receivers on the team’s opening drive of the quarter. Out of the shotgun, he threw a 20-yard touchdown score to Taysom Hill, a backup quarterback who also lines up as a halfback and receiver in certain packages.

With less than two minutes remaining, Brees went to work again, completing passes to Kamara, Cook and Thomas. A false-start penalty against the Saints drained 10 seconds off the clock, forcing them to kick a game-tying field goal on second-and-15 instead of having an additional chance to try for a game-winning touchdown.

The Vikings won the coin toss in overtime, and under N.F.L. rules, if a team scores a touchdown on the opening drive, the game ends. Cousins finally had his big moment. He hit Cook and Diggs — the hero two years ago — before seeing Thielen open downfield for a 43-yard completion. On the final play, Rudolph appeared to push his defender in the end zone before catching the winning pass. No offensive pass interference was called, and a video review confirmed the call on the field.