Stoll scored on a power play late in the first period to cut the Kings' deficit to 2-1 and Anze Kopitar got the equalizer while Los Angeles had a 5-on-3 advantage with 18 seconds left in the middle frame.
Quick stopped 20-of-22 shots and raised his career road shootout record to 8-1.
John Madden and Nick Schultz scored for the Wild. Backstrom turned aside 32-of-34 shots in the defeat.
Los Angeles captain Dustin Brown was given a game misconduct after the referee ruled he delivered an illegal blind-side hit to the head of Miettinen.
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