Slovak Cibulkova exited in straight sets 7-5 6-4 to Dutchwoman Kiki Bertens.
The 21-year-old Bertens earned a quarter-final against sixth seed Lucie Safarova, who beat Alize Cornet 6-2 4-6 6-2.
"Dominika's a good opponent and it was a really good match," Bertens said. "She was playing well so I got nervous the last game, but I'm happy I managed to keep serving well and I'm happy with the win."
Magdalena Rybarikova completed a day to forget for Slovakia as she retired hurt at 4-0 down in the first set against Errani.
Errani suffered a first round loss to Carla Suárez Navarro at the Australian Open but the top seed will get a chance for some revenge in the quarter-finals.
Errani said: "The next match is going to be interesting because I know Carla very well and she beat me at the Australian Open. I like this surface and I just want to keep playing my game, and we will see how it goes tomorrow."
There was also woe for eighth-seeded Czech Klara Zakopalova as she went down to Spaniard Carla Suarez Navarro.
Round two results
Kristina Mladenovic (France) beat Yanina Wickmayer (Belgium) 6-4 6-4
1-Sara Errani (Italy) beat Magdalena Rybarikova (Slovakia) 4-0 (Rybarikova retired)
6-Lucie Safarova (Czech Republic) beat Alize Cornet (France) 6-2 4-6 6-2
Carla Suarez Navarro (Spain) beat 8-Klara Zakopalova (Czech Republic) 6-4 6-3
Kiki Bertens (Netherlands) beat 4-Dominika Cibulkova (Slovakia) 7-5 6-4