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Sakkari saved four match points and trailed 5-2 in the second set but found her form in time to extend her run at the tournament, where she lost the final last year.
"I don't know where I found this power. Probably I have it deep inside. I'm a Spartan -- we don't forget that," the Greek 24-year-old said with a smile.
"I'm super happy. I was playing very bad tennis. I was not moving, not reacting at all. I really don't know how I managed to wake up."
A pep talk during the match from coach Tom Hill appeared to inspire Sakkari, who lost to Svitolina in their third-round meeting at Wimbledon last month.
Sakkari will next face Amanda Anisimova or Saisai Zheng on Saturday.
In another compelling quarter-final, second-seeded Aryna Sabalenka defeated Carla Suarez Navarro in a marathon 3-6 6-2 6-4 match earlier in the day.
Sabalenka rifled a forehand winner on match point to end the nearly two-hour affair and book her ticket to the last four as she hunts for her fourth career singles title.
Next up for Sabalenka is a semi-final with Donna Vekic, who put on a serving clinic to dispatch American Kristie Ahn 7-5 6-0.
Chris Froome (Stage 6, 2015)
Round 2, Kitzbuhel (2/3)