The 33-point halftime comeback surpassed the previous record of 32 set by the Buffalo Bills against the Houston Oilers in the 1992 playoffs.
The Colts piled on 17 unanswered points in the first quarter followed by another 16 in the next, leading the home side to be booed off US Bank Stadium.
Kevin O'Connell, who has now won 11 of his first 14 games as Minnesota head coach, admitted he spent the break challenging his team to claw back their first division title in five years.
"It was probably the most motivated I've been to challenge our players, so they just knew that regardless of the outcome today, that first half was not good enough," O'Connell said after the game.
"It will never be acceptable to have that type of performance, regardless of the outcome today.
"Nobody is happier to be NFC North champions. The way we earned it, I think, will stick with a lot of us for a long time, but there's still a lot to go back and make sure we rectify."
Something clicked halfway through the third quarter, with the Vikings then outscoring the Colts 39-6 to guarantee their playoffs spot.
Veteran quarterback Kirk Cousins threw for 417 yards and four touchdowns in the second half, with his side scoring five times over seven possessions and Greg Joseph completing the comeback as he kicked a 40-yard field goal in overtime.
The AFC North-leading Baltimore Ravens missed their star signal-caller, the injured Lamar Jackson, in a road loss to Deshaun Watson and the Cleveland Browns.
Watson, in his third game for the Browns after serving an 11-game NFL suspension for alleged sexual misconduct, threw a touchdown pass in his home debut as Cleveland defeated the Ravens 13-3 and kept their slim playoffs hopes alive.
And in the night's final game, the Buffalo Bills joyfully slid in the snow and into their fourth straight playoffs after Tyler Bass kicked a 25-yard field goal as time expired in their 32-29 AFC East win over the Miami Dolphins.