Babos struggled in the first set, but rallied to record a 4-6, 6-4, 6-2 victory in one hour and 54 minutes on the hardcourts at Sierra Madre Tennis Club.
Babos double-faulted three times in the opening set, but recovered to even the match in the second by winning 10-of-13 first serve points and breaking Cirstea twice.
Sixth-seeded Greta Arn was an easy second-round winner as well.
The Hungarian Arn piled up six service breaks in erasing Croatian qualifier Maria Abramovic 6-4, 6-2. Arn will face Russian Nina Bratchikova in Friday's quarterfinals.
Also recording wins on Thursday were Austrian Patricia Mayr-Achleitner and Mandy Minella of Luxembourg.
Mayr-Achleitner made quick work of Eva Birnerova of the Czech Republic with a 6-1, 6-3 victory in just one hour and 19 minutes. Mayr-Achleitner won 7-of-11 break points, while allowing Birnerova to capitalize on only 3-of-7 break opportunities.
This week's champion will claim $37,000.