On a windy day in the Hungarian capital, Scheepers got over the finishing line, 75 64, thanks to breaks in the final game of both sets.
"I'm happy to be in the quarters and to have made the transition to clay from grass fairly smoothly," Scheepers said. "Today's match was very close. Petra is such a good player, so I knew I had to take my chances when I got them. The conditions were very tricky, as it was quite windy, so I'm glad to have gotten through today's match."
Next Scheepers will play Danka Kovinic, after she became the first player from Montenegro to reach the last eight of a WTA event when she defeated Valeria Solovyeva, 61 36 61.
"Tomorrow will be another tough opponent, but I will just keep on trying to do the same things I did in my previous rounds," Scheepers added.
"I think I played really well and I'm really happy with my performance. The match was closer than the score makes it seem and the first set had lots of long rallies," Peer said. "I think the key moment was when I came back from 0-40 at 3-1 in the first set."
Facing her for a place in the semifinals will be Alexandra Cadantu, after her opponent, Andrea Hlavackova, retired with cramps at 46 75 40 down. Cadantu survived four match points at 5-3 down in the second set to advance to her third clay court quarterfinal of the season.
"Against Alexandra, I'll try and do the same things I did today. I'll try to be solid and keep a high level and try and play it one point at a time," Peer said.