The seven-time winner found her range at the start of the second set and, in front of the watching Duchess of Sussex, went on to beat the 18-year-old Slovenian 2-6 6-2 6-4.
After an error-strewn first set Williams rediscovered her consistency, and in doing so forced more mistakes from her young opponent.
A break in the first game settled the American, and a second break brought the match level.
The 23-time grand-slam winner, who has a doubles date with Andy Murray on Friday, was by now in the ascendancy.
Juvan, ranked 133 in the world, bravely clung on in the decider but a solitary break saw Williams safely through.
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