Mrazek's first game at pro hockey's highest level came at the age of 20 and he reacted like a veteran while his teammates gave him plenty of support. Detroit had been outscored 28-23 in its first nine games.
Daniel Cleary and Jakub Kindl gave the Red Wings a 2-0 lead with their first-period goals.
Alex Pietrangelo delivered the only score for St. Louis with 4:15 to go in the second period, but before that stanza was over the Red Wings had produce two more goals to take control.
Cory Emmerton scored with 2:27 left in the period and Damien Brunner came through with 56 seconds left. Valtteri Filppula finished the scoring with his third-period goal.
Brian Elliott was charged with all five goals on 29 shots.