Elliott turned aside all 24 shots he faced until Minnesota's Ryan Suter finally dented him with 8.2 seconds remaining in the game.
Elliott also notched an assist on a Jaden Schwartz goal in the second period.
T.J. Oshie added a score as St. Louis logged a seventh consecutive triumph over Minnesota and its fourth victory in five games overall.
Josh Harding stopped 21-of-23 shots for the Wild, who posted a sixth straight regulation loss for the first time in team history.
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