The win cut the Blues' lead in the first-round series to three games to one. The series shifts back to Minnesota for Game 5 on Saturday.
Blues goalie Jake Allen's first mistake in the series set up the first Minnesota goal by Coyle with 3:10 left in the first period. Allen was caught behind the net as he played the puck, and it went directly to Coyle, who fired it into the net for his second goal of the series.
The Wild never had a lead at any point in the first three games but were determined as they took the ice and it showed in the first period. At the time of Coyle's goal, they were outshooting the Blues 11-1.
Allen helped keep the Blues within one goal as they killed off consecutive power plays for the Wild midway through the second period but Hanzal's goal with 3:19 left put the Wild in front 2-0.
Hanzal was able to skate in between defensemen Alex Pietrangelo and Jay Bouwmeester and beat Allen for his fifth career playoff goal and first since 2012.
Dubnyk had to stop 12 in the third period.
Only four teams in NHL history have come back to win a playoff series after losing the first three games. The last team to do it was the Los Angeles Kings in 2014. The Blues were attempting to complete their first sweep of a playoff series since 2001.
Allen stopped 26 shots in the loss.
NOTES: C Erik Haula was not able to play for the Wild as he still has an upper-body injury, the result of a hard check into the boards in the second game of the series. He played in game three but C Joel Eriksson Ek made his first appearance in the series for game four. ... The Blues used the same lineup for the third consecutive game, but a change could be coming. C Paul Stastny, out since March 21 with what is believed to be a broken foot, has resumed practicing and might be able to play in game five on Saturday. ... C Alexander Steen did not practice for the last days for the Blues as he apparently deals with a lower-body injury, but he was still in the lineup Wednesday night.