The previous night, she had beaten Polina Kudermetova after matches were delayed due to rain. She was coming off having pulled out of the French Open due to illness.
Vekic had two break points against Rybakina in a match that lasted 2 hours and 14 minutes.
“I’m really happy to not only beat her but to be in the quarterfinals here,” Vekic after her match.
“I stayed in the moment. I was just trying to play the next point, the next game. She was serving unbelievable today.
She’s definitely, I think, the best server on the women’s tour right now. She was serving really big, so I was like, ‘Just stay with her; your chances will come,’ and they did.
“I was just trying to go for it, to be aggressive, which is not easy. In a way, I found myself lucky almost, every time I could get the return in.
I felt in a better position once I was in the rally, but to get her serve back was not easy.”
Vekic will next go up against Elina Avanayesan.
Elsewhere, Ons Jabeur who came in second to Rybakina at Wimbledon is already starting her grass-court season.