Shattenkirk hesitated after skating in on Minnesota goalie Josh Harding, then unleashed a wrist shot that went into the net high to the blocker side in the second round of the tie-breaker.
Jaroslav Halak wasn't dented in the shootout and finished with 29 saves for the Blues, who pulled into a first-place tie with Chicago atop the Central Division with 58 points.
Alex Pietrangelo and David Perron also found the net for St. Louis.
David McIntyre scored his first NHL goal and Jared Spurgeon also got a goal for the Wild, who lost for the fifth time in six games.
Harding made a career-high 47 saves for Minnesota.