With just over 60% of ballots counted in Monday's referendum, the first national vote since the small central African country was plunged into chaos in 1993, more than 90% were yes votes, officials said.
Vote counting started slowly because of logistical problems after the nationwide referendum on a constitution aimed at bringing an end to an 11-year civil war that has claimed 300,000 lives.
The independent national electoral commission (CENI) said early on Tuesday that the yes camp was ahead with 90.7% against 9.3% for the no supporters.
Turnout in Monday's referendum was over 87%, CENI chief Paul Ngarambe said.
"It's a tally that pleases us because it means that the Burundians have understood what is at stake in the referendum," he added.
"All the results are coming in, but we have problems with three provinces (out of 17) where the telephone's not working, so the results can't get through to Bujumbura," Ngarambe said.
President Domitien Ndayizeye said the referendum opened an "era of democracy" in his conflict-scarred country.
"This day will be marked in the annals of the history of the country," Ndayizeye said after casting his ballot in the capital.
"We were in a black period. Today, this is the era of democracy"
President Domitien Ndayizeye
"We lived in conflict since 1993. We were in a black period. Today, this is the era of democracy," added Ndayizeye, a Hutu who in February abandoned plans to modify the draft before the vote to allow him to stand for re-election this year.
The referendum is the first election to be held in this tiny central African nation since it plunged into chaos in 1993.
Voters are expected to endorse the constitution, which envisages a balanced power arrangement between majority Hutus and minority Tutsis who have dominated politics in the country of 7.1 million since independence from Belgium in 1962.
Final returns are to be announced on March 4.
Despite opposition from some in the Tutsi camp who feared the arrangement would erode their decades-long hold on power, the partial results indicated that at least some Tutsis had voted in favour of the constitution.
Under the constitution, Burundi's president will have a deputy from each of the ethnic groups, while 60% of the cabinet will be Hutu and 40% Tutsi.
"This day will be marked in the annals of the history of the country"
President Domitien Ndayizeye
Representation in the parliament, made up of a National Assembly and Senate, will be apportioned on a 50-50 basis, with Hutu and Tutsi parties required to field candidates from both ethnicities to reach the mix.
The constitution calls for the army and the police force to be equally split along ethnic lines.
Analysts predict that pro-referendum Hutu parties, to which about 85% of eligible voters belong, will easily overpower opposition from Tutsi parties, which account for only 14% of voters.
The referendum, postponed three times since last year, is the first step in a seven-tier election process, including legislative polls set for 22 April.
On that date, Burundians are to vote for members of parliament who will then elect a president at an as-yet undetermined date. The president must then appoint a government.