But the 27-year-old Dane found her footing and served out the match after a brief rain delay at 5-3 to clinch victory in 80 minutes.
When the action resumed, Svitolina, the world number five from Ukraine, regrouped and forced a decider after claiming the second set in 44 minutes during which both women struggled to hold serve.
In earlier quarter-final matches on Saturday, Simona Halep continued her quest to defend her Toronto title with a straight sets win, while Elina Svitolina needed three sets to book the last semi-final spot. "It's really hard to get to Cincinnati and there's no tickets, so I needed to get them".
Sixth-seeded Caroline Wozniacki recorded her first-ever victory over a reigning World No. 1 player on Friday with a 7-5, 6-7 (3), 6-4 win over Karolina Pliskova to advance to the semifinals of the Rogers Cup in Toronto. "I was just trying to be very focused on my game and just go for every shot".
"It was a very tough match for me mentally because it was the second time I was playing on this court today", said Svitolina. "But, you know, maybe that's also the key why it's been going so well, because I just walked in and I was more relaxed about the whole thing". I have many tournaments ahead. "I need some time to analyze this day (Saturday) and then move into final (Sunday), which is really soon".
"She's having a good year as well - injury free and playing really well", Svitolina said.
"And physically it was very tough because she gets to lots of balls".