Mikko Koivu's second score equalized the game 2 victory over the Blues.
Koivu and St. Louis' T.J. Oshie scored in the second round of the shootout, and Cullen began the third with a shot that trickled under Jaroslav Halak's pad.
Kevin Shattenkirk had a chance to keep the game going, but as he approached the net, he couldn't get the back of his blade on the puck. A sprawled Josh Harding stopped it with his pad, securing the fourth consecutive win for the Northwest Division-leading Wild.
Harding finished with 25 saves, while Halak turned aside 30 in regulation and overtime.
Oshie and David Backes scored for the Blues, who had won three in a row.
Oshie's wrister gave the Blues a 2-1 edge 6:07 into the third, and the Wild pressed to even the game down the stretch. They had several chances stymied, but broke through with 1:38 remaining while on the power play.
The puck was bouncing around during a scramble in front, and Koivu flipped it in the open net from the right side. Halak had denied several shots but couldn't keep track of every rebound, and Koivu's tally sent the game to overtime.
The Blues applied most of the pressure in overtime but couldn't cash in on three shots, including one Harding had to save at the buzzer.
Backes put the Blues on the board midway through the first period, but Koivu evened the score 2:37 into the second on a shorthanded tally.
Minnesota has won five of the last six matchups with St. Louis...Koivu had one goal through 19 games this season prior to Saturday.