The Vikings erased a 33-0 halftime deficit and grabbed the win on Greg Joseph's 40-yard field goal with two seconds left in the extra session.
The previous largest comeback in NFL history was the Buffalo Bills' rally from a 32-point deficit to beat the Houston Oilers in a Wild Card round playoff game in January, 1993.
"We just pulled off the biggest comeback in NFL history -- OK, I need a second," Minnesota quarterback Kirk Cousins said in an onfield interview with NFL Network in Minneapolis.
Cousins connected on three touchdown passes in the fourth quarter.
That included a 64-yard TD on a screen pass to Dalvin Cook that was followed by a two-point conversion pass to T.J. Hockenson that tied the score at 36-36 with 2:15 left in regulation.
The Vikings also got second-half touchdowns fro K.J. Osborn, C.J. Ham, Justin Jefferson and Adam Thielen while their often maligned defense held the Colts to three points.
Each team punted on its opening possession of overtime, the Vikings finally striking for the win on their second OT opportunity.
Minnesota had won the coin toss and had the first possession of overtime, but their drive stalled. Their defense then forced a punt and the Vikings had the ball at their own 10 yard line with 1:41 to play.
Cousins directed a six-play, 60-yard drive capped by Joseph's game-winner.
Cousins threw for a career-high 460 yards, including 417 in the second half and overtime. His four touchdowns helped overcome two interceptions.
The Colts had returned a blocked punt in the first quarter for a touchdown to seize an early 10-lead.
A Vikings fumble then gave Indianapolis the ball for a drive that resulted in another Colts TD.
Minnesota trailed 23-0 when cornerback Chandon Sullivan appeared to return a fumble for a touchdown, but the play had been whistled dead before the fumble.
The Colts then took a 30-0 lead when Julian Blackmon returned a Cousins interception 17 yards for a touchdown.
Indianapolis kicker Chase McLaughlin capped the first-half scoring with the third of his four field goals.
"It was an ugly first half," Cousins admitted.
He said there "was belief" in the Vikings locker room at halftime, when cornerback Patrick Peterson said "all we need is five touchdowns."
"I thought he was being sarcastic," Cousins admitted.
The Vikings have captured their first division title since 2017.
"It's really the goal every year when the season starts, it's to win your division, get a home playoff game and get in the dance," Cousins said.
"That's the first goal, we've been able to secure that. But we've got to play better."
