The 25-year-old Australian, who defeated Sachia Vickery and Dayeon Back in straight sets to reach the last eight, was beaten 4-6 6-3 6-2 by Emina Bektas on Friday in Seoul.
Birrell broke her American opponent twice, in the third and seventh games, en route to taking the first set.
But Bektas, who like Birrell was born in Germany, rebounded to take the second after breaking for 2-0 and powering on.
Birrell, the current world No.102 who is set to improve on her career-best ranking of 100, lost serve in the opening game of the third set. And she was broken again in the seventh before going on to lose the two-hour nine-minute contest.
The defeat equals Birrell's career-best last-eight appearance at Merida, Mexico in February.
In the day's other three quarter-finals, top seed and world No.4 Jessica Pegula faces fellow American Claire Liu.
Fourth-seeded Czech Marie Bouzkova will take on China's Yue Yuan, while Belgian Yanina Wickmayer and Polina Kudermetova of Russia wrap up the action.
At the Zhengzhou Open in China, Ons Jabeur won her opening match and advanced to the quarter-finals by defeating Lucia Bronzetti 6-3 7-6 (7-5).
Jabeur will next play Daria Kasatkina, who beat local wildcard Bai Zhuoxuan 6-2 6-4.
Laura Siegemund, who beat Liudmila Samsonova in three sets, will face Jasmine Paolini in the quarters.
And at the Hong Kong Open, Anastasia Pavlyuchenkova beat No.2 seed Beatriz Haddad-Maia 7-5 1-6 6-1 to advance to the last eight, where she will meet Katerina Siniakova.
Third seed Elise Mertens defeated qualifier Sofya Lansere 6-3 7-6 (7-5) to set up a match against the No.6 seed, Martina Trevisan, who beat Magdalena Frech.
Sara Sorribes Tormo put out the fourth seed and local favourite Wang Xinyu, 6-4 4-6 6-1, and will play the No.5 seed, Anna Blinkova, next.
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