The Ukrainian fought back from a disappointing second set to win 6-0 6-7 (5-7) 6-3 on Louis Armstrong Stadium.
Svitolina had been in control of the match before rain intervened on Tuesday, and she struggled for rhythm when play restarted on Wednesday.
"Today was a little bit of a mess in my head and it was very tough to keep the focus," she told Eurosport.
More to follow.
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Fourth seed Elina Svitolina overcame a scare to beat world number 42 Katerina Siniakova in the US Open first round.