Tight end Rudolph secured a four-yard pass from quarterback Kirk Cousins to cap a game-winning drive to open the additional period, silencing the majority of the crowd inside the Mercedes-Benz Superdome.
The victory means the Vikings – the sixth seeds in the NFC – will travel to San Francisco to face the 49ers next, aiming to cause a second shock result on their travels.
Once again in a postseason meeting between the teams, the Vikings found a way to get over the line despite squandering a 10-point lead in the fourth quarter.
Stefon Diggs scored on a 61-yard touchdown pass from Case Keenum to stun the Saints on the final play in January 2018, with the game later renamed the 'Minneapolis Miracle' due to the stunning finish.
This time Rudolph was the hero for Minnesota, though only after Cousins had set up the touchdown with a glorious 43-yard pass to Adam Thielen that moved the ball into the red zone.
Having also suffered a controversial exit at the hands of the Los Angeles Rams last year in the NFC Championship Game, New Orleans are left to come to terms with another heartbreaking play-off exit.
They had led 10-3 heading towards half-time, only to concede 10 points inside the final three minutes of the second quarter. Dan Bailey slotted over a field goal before the Vikings capitalised on a Drew Brees interception, Dalvin Cook going in from five yards out on the ground to make it 13-10.
The gap was extended further when the running back scored again in the third quarter, yet the Saints rallied. Brees connected with Taysom Hill for a 20-yard touchdown before Will Lutz levelled the score at 20-20, landing a 49-yard field goal with two seconds remaining in the fourth quarter. However, that would be the last time the home team had the ball.
After Thielen's catch, Cook failed twice to score on the ground from close range before Cousins went to the air again, picking out Rudolph in the back corner of the end zone.