Batal Tabagua, the head of Abkhazia's electoral commission, says 18 out of 35 legislative races have been decided.
A second-round runoff will be necessary in 17 districts where no candidate won an outright majority, apsny.ru reported. The runoffs will take place in two weeks.
Abkhaz Foreign Minister Sergei Shamba, speaking today to RFE/RL's North Caucasus Service, hailed the vote. "Abkhazia has made another step toward building a democratic, independent state," Shamba said. "The elections have been declared valid in most districts. In other districts, elections will be held once again."
Overall turnout across the breakaway province was around 47 percent, well above the 25 percent needed to make the poll legitimate.
Some 130,000 people were eligible to participate in the vote for Abkhazia's 35-seat parliament.
Of the 18 candidates elected so far, at least 11 represent the three political forces (United Abkhazia, Aytayra, and Amtsakhara) that support incumbent President Sergei Bagapsh, according to apsny.ru.
Five opposition candidates were also elected.
Such details are of little interest to Georgia. Tbilisi has refused to recognized Abkhazia's independence bid, launched in the early 1990s after a war with Georgia that left thousands dead and forced hundreds of thousands of Georgians to flee the Black Sea region.
Georgian President Mikheil Saakashvili said on March 4 that his country did not plan to "compromise with separatism."
"The number of people who were expelled from Abkhazia ranges between 400,000 to 500,000," Saakashvili said. "Given these circumstances, any attempt to legitimize this illegal act is unacceptable, inappropriate, and will never be recognized by either Georgia or the international community."
Georgian Prime Minister Zurab Noghaideli likewise rejected the vote, but said Georgia was ready to offer peace talks with Abkhazia that would lead toward "wide autonomy" for the region.
"These elections are not only illegal, but also immoral," Noghaideli said. "Therefore, as you are aware, these so-called elections cannot entail any moral or legal result. This is one thing that I wanted to say today. Nevertheless, we are offering peace talks to the de-facto leadership of Abkhazia -- without any postponement, delay or precondition. [These talks], in the long run, should become the foundation for a full and peaceful resolution of this conflict. We are offering wide autonomy to Abkhazia."
No country currently recognizes Abkhazia officially. However, it receives de facto support from Moscow, which keeps peacekeepers there.
A declaration issued on behalf of the EU by the bloc's current president, Germany, used the term elections in inverted commas. It stated that before any polls can be considered valid in Abkhazia, all refugees and internally displaced persons must be given "the right to a safe, secure and dignified return to their homes."
The EU also says it "fully supports" Georgia's territorial integrity within its internationally recognized borders.
Abkhaz officials expressed hope that the vote would boost independence efforts by demonstrate the region's commitment to free and fair electoral standards.
Foreign Minister Shamba said the ballot was a success despite what he called meddling by Tbilisi.
"The situation is normal even though the Georgian media and Georgian politicians have tried to draw a horrifying picture of Abkhazia," Shamba said. "But in reality it is not like that. None of the attempts by the Georgian leadership to start conflicts in Abkhazia among different religions, ethnic groups, or areas have succeeded here, because there are no such internal conflicts in Abkhazia."