The Bears scored 17 fourth-quarter points, capped by a 33-yard field goal by Austin Witmer as time expired, to seal the victory.
Missouri State was down 14-3 at halftime, with only a 47-yard try by Witmer.
In the third quarter, South Dakota took a 21-3 lead, but a Michael Cooper-Falls touchdown made it 21-10.
The Coyotes (1-5, 0-3) got a Kevin Robb 44-yard field goal to make it 24-10 shortly after.
With 5:45 left in the game, the Bears, who hadn't scored a point in the fourth quarter of a game all season, made it 24-17 with a Kierra Harris 60-yard touchdown pass to Julian Burton.
Anthony Grady came up with a fumble by USD quarterback Josh Vander Maten to give the Bears the ball.
Missouri State scored again, this time on a 25-yard strike from Harris to Nixa native Eric Christophel, to tie the game 24-24, with 1:45 remaining.
With zero time outs, Grady came up with a huge sack, his third. USD was forced to punt.
Missouri State drove the ball to the 16 yard line, and after two time outs called by USD to try and ice Witmer, he split the uprights for the 27-24 win.