The Colts (1-1) had surrendered a late 20-13 lead in the fourth quarter when Christian Ponder lead the Vikings on a nine-play, 47-yard drive that resulted in a game-tying, 6-yard TD toss to Kyle Randolph.
Indianapolis then got the ball back at its own 20 with 31 seconds remaining.
Andrew Luck began with a 20-yard pass to Donnie Avery (nine catches, 111 yards), then connected with Reggie Wayne for 20 more, putting the ball to the Minnesota 40.
After an offside call on Minnesota, Vinatieri handed Luck his first NFL win with a decisive field goal.
The NFL's first-overall 2012 draft choice went 20-of-31 through the air for 224 yards and two scores.
Ponder finished with 245 yards on 27-of-35 efficiency with two TDs and no interceptions for Minnesota (1-1).