McMichael scored with a wrist shot over Karel Vejmelka's glove for the deciding goal after Barrett Hayton missed the net on the previous attempt.
Jake Neighbours and Jimmy Snuggerud also scored in the shootout for the Blues (9-8-2), who have won two in a row. It was Hofer's second shutout this season.
Vincent Trocheck and Michael Carcone scored in the shootout for the Mammoth (10-8-2), who have lost four in a row and five of six (1-3-2), including 5-3 at the Vegas Golden Knights on Wednesday. Vejmelka recorded his second shutout this season and 10th in the NHL.
Mason McTavish appeared to open the scoring to give the Blues a 1-0 lead at 14:35 of the second period, but Utah challenged the play for offside, and the goal was overturned after a video review.
St. Louis forward Philip Broberg appeared to score the game-winner at 2:42 of overtime, but video review determined his wrist shot from the right face-off circle hit the crossbar after squeaking under Vejmelka's left arm.
The Blues were 0-for-5 on the power play. The Mammoth were 0-for-4.
