Elliott improved to 20-5-2 on the season, turning aside all 13 shots he saw as the Blues improved to 15-3-2 in their last 20 games.
Ryan Reaves and Barrett Jackman each notched their first goals of the season in the first period while Andy McDonald got his second of the year in the middle frame and Vladimir Sobotka also scored for St. Louis.
Minnesota, meanwhile, dropped its seventh straight game, mustering only three shots in each of the final two periods.
Josh Harding allowed four goals on 34 shots for the Wild.
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