Bogdan 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, 6-4 and Australia's Kimberly Birrell was leading 3-6, 7-5, 4-1 when third seed Elise Mertens of Belgium retired.
Seventh-seeded Diane Parry avoided the upset bug with a 6-4, 6-2 defeat of Belgium's Greet Minnen.
Ningbo Open
Fourth-seeded Barbora Krejcikova of the Czech Republic moved into the quarterfinals in China with a 7-6, 6-3 victory against home hopeful Ma Ye-Xin.
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, 7-5 while seventh seed Anna Kalinskaya of Russia beat China's Yun Yue 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.