The Palestinian Election Committee said Fatah won control of 52 of 84 municipal councils across the West Bank and the Gaza Strip and Hamas won 24.
Smaller factions took four councils, with four municipalities undecided in elections widely seen as an indicator of public sentiment ahead of a parliamentary election in July.
Official results are expected on Sunday.
Speaking to Aljazeera, Abbas praised the atmosphere under which the elections were held, describing them as "transparent and democratic".
An electoral commission official said there was 70% voter turnout in the West Bank and 80% in the Gaza Strip in the elections to choose 906 municipal council representatives from among 2519 candidates, including 399 women.
Hamas, which made inroads in several key population centres, contested the unofficial results. A Hamas statement said the group won in 34 municipalities including in Qalqaliya, out of 68 in the West Bank. In the Gaza Strip, Hamas said it won four municipalities out of seven, including the Rafah municipality.
80% of voters turned out to cast
ballots in the Gaza Strip
The group urged the media to be cautious in reporting the outcome of the election and warned of violations.
Hamas leader Shaikh Hasan Yousef said in a statement obtained by Aljazeera: "A group of masked men have stormed the election counting centres in Attarah area and burned the ballot boxes after the announcement of the victory of the al-Wafaa and Construction list, which supports Hamas."
Yousef slammed what he termed a "passive and indifferent attitude" of the international observers who did not meet Hamas candidates to follow up complaints about repeated violations.
Stepping up reforms
Hamas' support is still growing while Fatah has seen its popularity slip amid corruption allegations.
But political analysts said Fatah had fought back by stepping up internal reforms aimed at giving a younger generation of activists a stronger voice in the mainstream faction.
It did well in recent student elections in several Palestinian universities and fielded reformist candidates for the municipal vote.
The election came against the backdrop of a fragile ceasefire with Israel engineered by Abbas that has raised hopes of reviving Middle East peacemaking after four and a half years of Palestinian uprising.