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In her third attempt of six, Van Dyk launched a 63.93m heave, landing the javelin just short of the personal best of 64.22m, which she had set in the qualifying earlier this week.
Van Dyk finished behind Japan's Haruka Kitaguchi, who secured gold with a first-round throw of 65.80m with Czechia's Nikola Ogrodnikova taking bronze.
The 26-year-old Van Dyk became the second South African to medal in javelin, replicating Sunette Viljoen's silver at the 2016 Rio Olympics.
President Cyril Ramaphosa was quick to congratulate Van Dyk, writing: "Well done and thanks for making Team South Africa and all of us proud."
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