Minnesota, who trailed 36-7 with under five minutes to play in the third quarter, put the finishing touches on the comeback when Greg Joseph nailed a 40-yard field goal with three seconds left in the extra period.
With the win, the Vikings improved to 11-3 and secured the NFC North Division title - their first since 2017 - with three games left to play in the 2022 regular season.
The previous record for the biggest comeback in NFL history was set by the Buffalo Bills in 1993 when they erased a 35-3 third-quarter deficit to beat the Houston Oilers 41-38 in overtime of a wild card playoff game.
Indianapolis built a 17-0 first-quarter lead after a field goal, a blocked punt that JoJo Domann returned for a score and a one-yard Matt Ryan touchdown pass to Deon Jackson.
They padded their lead in the second quarter with another three field goals and a 17-yard interception return that Julian Blackmon returned for a touchdown.
But the Vikings cut into the deficit with a pair of third-quarter touchdowns, the second of which came after Chase McLaughlin put the Colts ahead 36-7 when he kicked a 52-yard field goal.
Minnesota then added another three touchdowns in the fourth quarter, tying the game with about two minutes left when they converted a two-point conversion after Dalvin Cook took a screen pass from Kirk Cousins and ran it 64 yards for the score.
The victory also handed Indianapolis quarterback Ryan the dubious distinction of having been on the losing end of both the largest comeback in Super Bowl history and NFL history.
Ryan and his former team, the Atlanta Falcons, led the New England Patriots 28-3 in the third quarter of the Super Bowl in February 2017 before the Patriots pulled off a comeback for the ages and won 34-28.
