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The 32-year-old beat Austria’s Michaela Polleres in the Repechage in the -70kg category.
Conway previously won bronze at the 2014 Commonwealth Games and 2016 Rio Olympics.
Her hopes of gold ended with a semi-final defeat to France's Marie-Eve Gahie, but she quickly bounced back to floor Polleres and emerge with another medal. Gahie went on to win gold.
Conway was GB’s flag bearer at the opening ceremony.
‘Beautiful stuff!’ – Conway floors rival to win bronze
Round 13, Zurich
Semi-final 2, Kitzbuhel (2/2)