The 92nd-ranked Dolonc broke the Slovak for a 4-2 lead in the third set and then saved a break point while serving for the match.
Earlier, Magdalena Rybarikova of Slovakia ousted Russian wild-card entry Ksenia Pervak 7-5, 6-2.
In the first round of the men's tournament, Paolo Lorenzi advanced when Filippo Volandri retired with a right hip strain. Lorenzi was leading 3-6, 7-5, 4-1.
Volandri served for the match in the second set while leading 6-3, 5-4, but Lorenzi broke him to level.
"He had one match point so I was lucky," Lorenzi said. "It wasn't easy because we are very good friends. This morning we made the warm up together."
In an all-Italian second round, Lorenzi will face defending champion Andreas Seppi, who had a first-round bye.