Zvonareva, who won a tournament in the Czech Republic two weeks ago, extended her personal winning streak to six matches with a 6-3, 6-3 win over Ai Sugiyama. She's advanced to at least the quarterfinals in eight of her nine tournaments this year.
Zvonareva leads the WTA with 32 match wins this year.
Szavay posted an even more lopsided 6-1, 6-1 win in her match with Zi Yan.
Also winning was 17th-seeded Francesca Schiavone, who defeated Lucie Safarova 6-4, 7-5.
Tournament officials said Dinara Safina, the Italian Open runner-up two years ago, had withdrawn from the 2008 event because of a back injury. Safina beat world No. 1 Justin Henin and No. 6-ranked Serena Williams last week en route to the championship of the German Open, which she wrapped up Sunday.