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"I'm happy to have made the transition to clay from grass fairly smoothly," said Scheepers, the sixth seed.
"Petra is such a good player, so I knew I had to take my chances when I got them."
Scheepers' next opponent will be Danka Kovinic of Montenegro, who beat Valeria Solovyeva of Russia 6-1, 3-6, 6-1.
Shahar Peer of Israel knocked out eighth-seeded Anna Tatishvili of Georgia 6-1, 6-1, leaving just three seeded players in the last eight.
Peer, who won her last WTA tournament in 2009, will face Romania's Alexandra Cadantu.
Cadantu defeated Andrea Hlavackova of the Czech Republic, who retired due to cramping with the Romanian leading 4-6, 7-5, 4-0.
Results from the day four of the WTA Budapest Grand Prix (x denotes seeding):
Second round:
Danka Kovinic (MNE) bt Valeria Solovieva (RUS) 6-1, 3-6, 6-1
Chanelle Scheepers (RSA x6) bt Petra Martic (CRO) 7-5, 6-4
Shahar Peer (ISR) bt Anna Tatishvili (GEO x8) 6-1, 6-1
Alexandra Cadantu (ROM) bt Andrea Hlavackova (CZE) 4-6, 7-5, 4-0 - retired
