The top seed, who has an opportunity to secure her place at the WTA Finals a week early, was in trouble as she trailed Chinese star Wang Qiang 6-2 5-2.
Wang was set to serve for the match when rain stopped play and the pair will return to the court ahead of the semi-finals on Saturday, with Svitolina facing a huge uphill battle if she is to add points to her bid to reach Singapore.
Garbine Muguruza awaits the winner of that tie after she reached her first semi-final since the French Open in June.
The two-time grand slam champion defeat Luksika Kumkhum 6-2 7-5 and said: “I’m very happy with the way I’m playing so far. Luksika played great and made it a real battle out there.”
Muguruza has won three consecutive matches for the first time since her run to the last four at Roland Garros, not dropping a single set this week and breaking a streak of seven straight exits in the second or third round.
On the other side of the draw, Shuai Zhang dumped out seventh seed Daria Gavrilova 6-1 6-3 to secure a last-four meeting with Dayana Yastremska, who defeated Kristina Kucova 7-6 (8-6) 6-2.
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