Bogdan, who reached the third round of this year's French Open, overcame a 3-1 deficit in the tiebreak to finish in just under 2 hours, setting up a match with unseeded Suzan Lamens of the Netherlands.
It was a day of upsets, as Japan's Sara Saito defeated fifth seed Elina Avensyan, of Armenia, 7-6 (7-2) 6-4 and Australia's Kimberly Birrell was leading 3-6 7-5 4-1 when No.3 Elise Mertens, of Belgium, retired.Â
Seventh-seeded Diane Parry avoided the upset bug with a 6-4 6-2 defeat of Belgium's Greet Minnen.
Meanwhile fourth-seeded Czech Barbora Krejcikova moved into the quarter-finals in China with a 7-6 (7-5), 6-3 victory against home hopeful Ye-Xin Ma at the Ningbo Open
Ma had her chances, leading 6-5 in the opening set and 5-3 in the tiebreak. Krejcikova will meet Russia's Mirra Andreeva in the final eight.
Sixth seed Beatriz Haddad Maia held off England's Katie Boulter 7-6 (7-2) 7-5 and No.7 Anna Kalinskaya, of Russia, beat China's Yue Yuan 6-3 6-3.
Karolina Muchova, of the Czech Republic,needed only 77 minutes to complete a 6-2 6-1 victory over Romania's Jaqueline Cristian.
