The results released on Friday showed Hamas making strong inroads in key urban centres in the West Bank and Gaza, an indicator it could do well in this summer's parliamentary ballot, possibly complicating Abbas' peace efforts.
Fatah won control of 52 of the 84 municipal councils being contested in the West Bank and Gaza in Thursday's vote, while Hamas won 24, the Palestinian Election Committee said. Smaller factions took four councils, and four were undecided.
Hamas disputed the results, saying it had captured councils in 34 towns and villages.
A prominent leader in the Hamas Movement, Shaikh Hassan Yousef, said in a statement - a copy of which was obtained by Aljazeera - that a group of masked men had stormed the election centres in Attarah area and burned the ballot boxes after Hamas affiliates were declared winners.
"We are continuing our policy ... to reorganise all the institutions of the Palestinian Authority"
Shaikh Yousef slammed what he termed the "passive and indifferent attitude" of the international observers who did not even seek to meet Hamas candidates to follow up their complaints regarding violations of the electoral process.
A team of European observers said that except for a few minor problems, the elections had been fair and democratic, meeting a condition set by international aid donors.
Earlier, Mahmoud al-Zahar, a Hamas leader in Gaza, said many of the independent candidates who won council seats in the West Bank actually belonged to Hamas but chose not to run as members of the group for security reasons.
The election was fought against the backdrop of a fragile three-month-old ceasefire with Israel engineered by Abbas, a US favourite, a deal that has raised hopes of reviving Middle East peacemaking following four and a half years of violence.
Fatah is committed to a two-state solution with Israel, while Hamas, the driving force behind a bombing campaign against Israelis during the Palestinian uprising, is dedicated to the destruction of the Jewish state.
Fatah, founded by the late Yasser Arafat and long the dominant mainstream political faction, had seen its popularity slip amid allegations of corruption and mismanagement.
Hamas, which boycotted previous polls, posed an electoral challenge to Fatah after gaining street credibility for its
fight against Israel, its piety and charity work.
Fatah officials put the best face on the election outcome, which they hope will bolster Abbas' reform programme.
"We are continuing our policy ... to reorganise all the institutions of the Palestinian Authority," said Fatah's Abdallah al-Ifranji, a close Abbas aide, who said the election results indicated public approval of reform efforts.
Fatah gunmen fired volleys in the air in celebration on Thursday night.
Hamas leader Mahmoud al Zahra
(R) disputed some of the results
Khaled al-Qawasmeh, minister of local government, said Fatah won between 55% and 60% of the popular vote compared with 30% to 35% for Hamas.
Despite the results, some Fatah officials fear a heavy defeat by Hamas in parliamentary polls due in July, and Ifranji said the possibility of delaying the polls was being discussed.
Israel's pullout from the Gaza Strip, set to begin on 20 July, would be the main factor in deciding whether to postpone the vote, he said. Israeli officials have said the withdrawal may be delayed until mid-August to avoid a clash with a Jewish religious mourning period.
Hamas showed its strength on Thursday by winning control of municipal councils in several of the largest urban areas, including Rafah, Beit Lahiya and Bureij in the Gaza Strip and Qalqilya in the West Bank.
Final official results were expected early on Sunday. More than 2500 candidates vied for the council seats and turnout was high - 80% in Gaza and 70% in the West Bank, officials said. About 400,000 Palestinians were eligible to vote.
Palestinian President Abbas on Aljazeera praised the atmosphere under which the municipal elections were held, describing them as "transparent and democratic".