Davis had a relatively easy time with Cirstea, her 61 63 triumph putting her into her second WTA quarterfinal, her first occurring last September in Québec City.
"I was playing really well, waiting for my chances, playing my game," Davis said. "I served well, I returned well. I did everything pretty well.
"Getting deep into the tournament, it feels good."
From 4-4 in the first set of her second round match against Romania's Simona Halep on Tuesday, Stephens reeled off eight straight games for a 64 60 victory.
"I was up 2-0, and then she kind of settled down and started making a lot of balls and I got frustrated with myself," Stephens said. "Then I calmed down, served well and improved my attitude.
"I did fitness all fall because I didn't play any events. It was really cold out, so my nose was stuffed up and I was breathing really hard, so I was like, 'Oh my God, people are going to think I'm out of shape.' But I'm really pleased with the way I played and my fitness was good, so I'm happy with that."
Stephens talked about her upcoming showdown with Davis.
"It will be a rematch of our 18s hardcourt USTA Nationals match, so I'm excited to play her," Stephens said. "Out of all the times that we've been at the same tournament, we've never played at an actual WTA tournament, so it should be fun."
Also advancing to the quarterfinals were Elena Vesnina and defending champion Mona Barthel. Vesnina was a 46 62 61 winner over fourth-seeded Yaroslava Shvedova a week after also beating the Kazakh in the second round of Auckland. Barthel edged Chanelle Scheepers, 76(3) 75, to improve to 10-0 in Hobart, including qualifying matches.
"I had a good tournament last week in Auckland, playing four good matches and making the semis," Barthel said. "I worked really hard in the off-season and I feel like I improved a lot and I'm playing much better than last year. I'm just really excited for the new season, and we'll see how it goes."