Dom Parsons' men's skeleton bronze on Friday was Britain's first medal of the Games - and Atkin, who was third in the 2017 World Championships, scored 84.60 with her final run to claim another.
Sarah Hoefflin of Switzerland won with 91.20 as Mathilde Gremaud of Switzerland finished with silver with 88.00.
Atkin, who is based in Park City, Utah, and has British and Malaysian parents, qualified in fourth after an error on her first run, and Katie Summerhayes advanced in 10th.
Summerhayes, world silver medallist in 2015, scored 71.40 for her second run of the three-run final, good enough for seventh.
Atkin was awarded 79.40 for her second run to move to third entering the final run, which she began knowing she needed to improve to get on the podium. And she did for an historic British medal.
Alain Baxter was stripped of slalom bronze in 2002 for an anti-doping rule violation relating to a common cold medicine, while Jenny Jones won Britain's first Olympic medal on snow with bronze in the women's snowboard slopestyle event four years ago in Sochi.
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