The Belgrade-based Center for Free Elections and Democracy said their sample count after polls closed in the two-day vote indicated that 96 percent of those who participated in the referendum supported the draft charter.
At least 50 percent of the country's 6.6 million voters had to participate for the results to be valid and the group estimated turnout at 53.3 percent.
The final result is expected on Monday.
Serbian Prime Minister Vojislav Kostunica congratulated the country on its new constitution.
"This is a great moment for Serbia," Kostunica told Serbian television. "This is a historic moment, a beginning of a new era for Serbia."
The referendum had been strongly condemned by the ethnic Albanians, who have long boycotted any ballot under Serb auspices.
Western diplomats have warned that only the international negotiations can decide on Kosovo's future, but Belgrade politicians claim adopting the new constitution would bolster their position in the talks.
Serbia's opposition Liberal Party claimed there was "massive fraud" at polling stations in the final hours of voting, with individuals allegedly voting several times and without identification papers.
The charter's key point declares Kosovo an "integral part of Serbia" despite ongoing U.N.-brokered talks on the province's future status.
Serbs in Kosovo — where the 2 million strong majority ethnic Albanian population was not even invited by Belgrade to vote — began celebrating even before official results were announced.
Hundreds gathered Sunday night in the Serb-populated northern Kosovo town of Kosovska Mitrovica, waving banners, cheering and shouting "Kosovo is the heart of Serbia" as NATO peacekeepers watched on.
Kostunica had earlier Sunday issued a final appeal on all voters "who hold Serbia in their hearts" to head to the polls and back the new constitution, warning of "unforeseeable consequences" if it is not voted in.
"Citizens, go out and vote 'yes,' for a better life for everyone," Serbian President Boris Tadic said in his own eleventh-hour plea.
Serbia saw a massive government campaign to say `yes' to the document, with ads flashing at the top of television screens Sunday on the state-run broadcaster calling on people to vote in favor of the draft constitution.
The 206-article draft — offered as consolidating democracy and the rule of law in the Balkan country — has focused on the preamble seeking to prevent a possible secession of the disputed Kosovo.
The province's independence-seeking ethnic Albanians form 90 percent of the population in Kosovo, which has been under U.N. administration since 1999, when U.S.-led NATO air strikes halted a Serb crackdown on the separatists.
Serbs cherish Kosovo — which today is home to a dwindling Serb community of 100,000 — as their historic heartland.
Government opponents and nongovernment organizations have criticized the charter as hastily drafted and flawed on issues such as independence of the judiciary, equal rights for minorities and autonomy for local governments.
The pro-Western Tadic has defended the charter as a break with the era of the late autocrat Slobodan Milosevic. Tadic also said the new constitution could propel Serbia toward coveted membership in the EU and NATO.
The need for a new constitution arose in June after Montenegro — Serbia's last partner from the former Yugoslav federation — declared independence and left Serbia on its own for the first time since 1918.