The 25-year-old Australian recorded a 68-minute 6-0 6-1 demolition to set up a last-eight clash with American Emina Bektas, who earlier on Thursday scored a 6-3 6-4 win over another South Korean in Su Jeong Jang.
After winning her opening-round match in straight sets by defeating American Sachia Vickery, Birrell took little time in claiming the opening eight games of her second-round contest in Seoul.
The current world No.102, who has a career-best ranking of 100, broke again in the fifth and seventh games of the second set to wrap up one of her most convincing victories on tour.
The win matches Birrell's career-best singles result at WTA level, having made her first quarter-final appearance at Merida, Mexico in February.
Back, a 21-year-old ranked No.569, had stunned the tournament's second seed, world No.13 Jelena Ostapenko, in the opening round.
Elsewhere, fourth seed Marie Bouzkova was leading 6-0 1-0 when her German opponent Eva Lys retired with injury. The Czech will now face China's Yue Yuan in the quarters.
In the day's final singles clash, before Birrell teamed up with Bektas in a losing doubles quarter-final match, top seed and world No.4 Jessica Pegula saw off teenager Ashlyn Krueger 6-3 6-1.
Pegula will take on another American in Claire Liu in the last eight.
Meanwhile, Ons Jabeur won her opening match at the Zhengzhou Open and advanced to the quarter-finals by defeating Lucia Bronzetti 6-3 7-6 (7-5).
Jabeur will next play Daria Kasatkina, who beat wild card Bai Zhuoxuan 6-2 6-4.
Laura Siegemund, who beat Liudmila Samsonova in three sets, will face Jasmine Paolini in the quarters.
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. Pavlyuchenkova will meet Katerina Siniakova next.
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 in three sets.
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.