Szavay, ranked 14th, committed several double faults in the first set and often appeared to lose her concentration.
"At Wimbledon, I didn't have a tenth of the excitement I felt here,'' Szavay said. "Playing before the home crowd usually helps players but now this backfired. I felt too much pressure.''
Fifth-seeded Lucie Safarova of the Czech Republic also had an early exit, losing to Anna-Lena Groenefeld of Germany 6-1, 6-4.
"I knew when I went on court that I would have a chance. It is a surprise but I'm really happy,'' Groenefeld said. "I think I served well today and played very aggressively from the baseline.''
Groenefeld will face second-seeded Alize Cornet of France in the quarterfinals. Cornet defeated Magdalena Rybarikova of Slovakia 6-2, 6-2.
Also, Karolina Sprem of Croatia defeated Tzipora Obziler of Israel, 6-2, 6-1.
The other four second-round matches are scheduled for Thursday at the Romai Tennis Academy.
WIE HAPPY WITH OPENING ROUND
Mauresmo holds off sixth-seeded Alize Cornet
Pirés: Raúl-Aragonés Is Like Me And Domenech
