Turnout on Monday was not as high as last month, but "acceptable"
The FDD won 58%, with President Domitien Ndayizeye's Frodebu party coming second with 22%, they say.
The FDD has said it wants to tackle poverty and promote reconciliation after 12 years of ethnic conflict between the Tutsi-led army and rebels.
The MPs choose a new president in August, tipped to be FDD leader Pierre Nkurunziza, an ex-university lecturer.
Correspondents say the FDD has more Tutsi officials than other mainly Hutu parties and Burundians hope this will mean they can reduce tension between the two communities.
NATIONAL ASSEMBLY FDD - 59 seats Frodebu - 24 seats Uprona - 10 seats Others - 7 seats
The last time a Hutu won presidential elections, in 1993, the Tutsi-led army staged a coup but correspondents say this is not likely now, as African leaders would not tolerate a military takeover.
No intimidation
The polling was largely peaceful with turnout put at 65%.
The main party representing the Tutsi minority, Uprona, came in third place.
Former rebel leader Pierre Nkurunziza (l) is expected to be Burundi's next leader
Reports of fraud were "not sufficient... to affect the overall result", she told a press conference.
There had also been no complaints of intimidation, which was encouraging, she said.
Astere Kana, a spokesman for Burundi's independent electoral commission, said the voting had "ended peacefully".
Many voters had feared attacks by the FNL, Burundi's last active Hutu rebel group.
During last month's polls, it was blamed for a series of shootings and grenade attacks.
Power shared
Polling stations were guarded on Monday by Burundian security forces and UN troops.
A Burundian soldier was slightly injured when a hand grenade exploded in the capital, Bujumbura, as ballots were being counted, Burundi's deputy police chief Col Helmenegilde Mimenya told the AFP news agency.
The BBC's Rob Walker in Bujumbura says the elections mark a fundamental shift of power in Burundi.
For much of the time since independence, control of the state has been in the hands of a narrow Tutsi elite.
Now, after a peace process lasting five years, Burundians are finally voting to a elect a new assembly in which power will be shared.
Forty per cent of the seats in the 100-member assembly are reserved for the Tutsi minority and 60% for Hutus.
180px  
One of Burundi's most popular radio stations, "Radio Publique Africain", has been allowed to re-open. 

RPA was  forcibly shut down on Friday July 22, allegedly on the orders of President Domitien Ndayizeye.

The move came just a day after RPA resumed broadcasting, having  observed an earlier two-day ban, following a disagreement with Burundi's "National Communications Council" (CNC). 

The CNC had accused RPA of giving disproportionate coverage to the opposition FDD party, who heavily defeated Ndayizeye's FRODEBU party in last month's national elections. Ndayizeye is due to relinquish power in August. 

The radio station's Director, Alexis Sinduhije, had denied the CNC's charges. President Ndayizeye was quoted by BBC monitoring as saying that he had decided to close the radio station to "safeguard the integrity of state institutions."

The closure of RPA was condemned in Burundi, where the radio station, founded in 2001 to promote reconciliation between the Hutu and Tutsi ethnic groups, has a wide following. The US-based "Committee to Protect Journalists" led international condemnation of the ban, which Burundi's "Collectif des Professionels des Médias de la Diaspora Burundaise" described as an "abuse of power", setting up an online petition in protest. 

"The National Council of Communication judged that RPA has respected council sanctions, but we have also allowed the radio to reopen because a lot of people requested it," CNC chairman Jean Pierre Manda told Reuters, earlier this week.

Manda has now resigned, although he denies that the move is connected with the row over RPA, and cites "personal reasons". The Committee to Protect Journalists reports that the CNC is to be reformed, with Radio Publique Africain's Deputy Director sitting on the board in future. 

There is some confusion over when the ban was lifted, with Reporters Sans Frontieres and the Committee to Protect Journalists reporting that it remained in place until Wednesday this week, while Reuters claims that the ban was lifted on Monday. 

RPA Director Alexis Sinduhije has told Reuters that some of the radio station's material has been destroyed by  the Burundian police, and suggested that broadcasting will not resume until a damage inventory has been taken.

This week's row is merely the latest manifestation of a long-running personal dispute between President Ndayizeye and the RPA Director Alexis Sinduhije, says journalist Gratien Rukindikiza, writing on the Burundian news website Burundi.news.free.fr.  Rukindikiza traces the root of the affair to a telephone call last year in which Ndayizeye tried, unsuccessfully, to persuade Sinduhije to endorse his campaign, through Radio Publique Africain, to amend Burundi's constitution. The amendment would have allowed Ndayizeye to seek a second term in power, in contravention of a 2003 peace agreement.