Kanepi, who beat Suarez Navarro for the Portugal Open title in 2012, ousted fifth-seeded Sorana Cirstea with a dominant 6-1, 6-2 win. Kanepi won more than 60 percent of the points in the match and saved all four break points she faced.
She took 57 percent of the points off Cirstea's serve and finished off five breaks.
Suarez Navarro, seeded fourth this year, stopped Yanina Wickmayer 6-4, 6-3. She lost all three breaks points Wickmayer had but had six breaks of her own.
While they can't meet in the finals this year, if Kanepi and Suarez Navarro both win their next matches, they would square off in the semifinals.
No. 3-seeded Anastasia Pavlyuchenkova defeated Estrella Cabeza Candela 6-3, 7-6 (7-4) and seventh-seeded Elena Vesnina beat Monica Niculescu 6-4, 6-4 and are the only other seeded players to reach the quarterfinals.
Other second-round results saw Romina Oprandi get past Peng Shuai 6-1, 6-3; Svetlana Kuznetsova oust Galina Voskoboeva 6-4, 6-4; Monica Puig take out Francesca Schiavone 6-3, 6-2; and Ayumi Morita beat Urszula Radwanska 6-2, 6-3.