The sixth-seeded Stephens matched her biggest win in terms of ranking in upsetting World No.3 Maria Sharapova in the second round last week. However, in the third round against Jelena Jankovic she was two points from victory but was unable to close it out.
"I had some ups and downs but overall I thought I played pretty well," Stephens said. "She obviously is a good player. I'm just happy to get through."
Stephens commented on the US Open, which starts next week.
"The US Open is something every American looks forward to," Stephens said. "I'm just really excited to go to New York. Hopefully I'm going to do my best in New Haven before heading there."
Carla Suárez Navarro's 75 63 defeat of Vinci wasn't a huge surprise, though, as the Spaniard is a Top 20 player herself and is now 4-1 in their series.
"It was a very good match," Suárez Navarro said. "Vinci came from Cincinnati and was a little bit tired, I think, but it was still tough. She is playing very good this year. It was a good win for me."
The competition stays tough for Suárez Navarro in the second round, where she will take on Simona Halep, a 62 61 winner over Daniela Hantuchova. The Romanian's banner season continued last week with her first hardcourt WTA quarterfinal showing in Cincinnati.
As for Cibulkova, she was up 63 21 on Klara Zakopalova before the Czech rallied for a 26 63 63 win - her third straight in their series. Promising Puerto Rican teen Monica Puig will be Zakopalova's second round opponent.