Firas Iari, the director-general of the local electoral commission, said Fatah had won 50 and Hamas 28 of the municipal councils in the West Bank and Gaza Strip, where voting took place on Thursday.
"Fatah got 59.9% of the votes cast, against 33.3% for Hamas," Iari said.
The Palestinian Centre for Policy and Survey Research earlier said Fatah appeared to win six of 14 municipal councils where it conducted exit polls in the occupied West Bank and Gaza Strip.
Palestinians were voting in municipal elections that had shaped up as a face-off between Abbas' Fatah movement and Hamas.
More than 2500 candidates were vying for seats on 84 municipal councils across the Israeli-occupied West Bank and Gaza Strip in a test of sentiment over the halting pace of political and security reforms pledged by Abbas.
"I think this election is a sort of referendum on Abu Mazen's policies," voter Amal Salahat said in the West Bank town of Bethlehem at the 7am (0400 GMT) start to polling, referring to Abbas by his popular nickname.
There were 406,000 eligible
voters in 84 localities
The vote sheds light on prospects for Fatah and its main rival, Hamas, in a July parliamentary election that Abbas hopes will strengthen his mandate for peace negotiations with Israel.
According to the elections-steering committee, 2519 candidates, including 399 women, contested 906 seats.
Unlike the first phase of local elections in December, the present elections covered some large towns such as Rafah at the southern tip of the Gaza Strip, Bethlehem, Bait Sahur, Bait Jala, Salfit and Qalqilya.
In all, the number of eligible voters in the 84 localities was 406,000, two-thirds of them in the West Bank and the others in the Gaza Strip.
Balloting proceeded against the backdrop of a fragile ceasefire with Israel engineered by Abbas and seen as crucial by US-led mediators to reviving a road map peace process towards Palestinian statehood in the West Bank and Gaza.
But tensions flared anew on Wednesday when Israeli troops shot dead two teenage Palestinian protesters and the Israeli government indefinitely suspended promised military pullbacks in the West Bank, citing Abbas' slowness to disarm fighters.
Analysts expected neck-and-neck races for the municipal councils as Fatah seeks to recover from a thrashing by Hamas in a January round of voting in Gaza.
Polling in Gaza got under way in a festive atmosphere as flag-waving activists for various factions erected tents and greeted voters flocking early to ballot stations.
Israeli troops killed two teenage
Palestinians on Wednesday
Hamas supporters sported green scarves while backers of the smaller leftist group, the Popular Front for the Liberation of Palestine, were clad in red.
Hamas was running on the slogan "partners in blood, partners in decision-making". Some voters said they wanted more power-sharing after decades of Fatah domination.
"I prefer that two factions rather than one be in charge for the sake of balance. People want the new council to end corruption and get rid of unqualified officials," Khalil al-Ashqar, 51, said as he voted in the Gaza town of Bait Lahya.
A recent poll showed that backing for Fatah slipped to 36% in March from 40% late last year, extending a downward trend, while Hamas support rose to 25% from 18%.
But 75% of Palestinians were satisfied with steps Abbas has taken to revive a Middle East peace process frozen since 2000, and 62% thought he was serious about fighting corruption, the poll showed.
"I prefer that two factions rather than one be in charge for the sake of balance"
Abbas, who took office in January, last month ordered a major shake-up of security forces that purged many loyalists of late president Yasser Arafat. But there have been little tangible results.
Fatah has also stepped up campaigning to counter gains by Hamas, particularly in the Gaza border town of Rafah - the largest municipality involved in Thursday's vote and a resistance stronghold where Hamas commands strong support.
Hamas spokesman in the West Bank Hasan Yousuf suggested that the PA chose to hold the elections in areas considered traditional Fatah strongholds.
"They (the PA) have refrained from holding elections in the large towns such as Nablus, Hebron, Gaza and Ram Allah. The reason is obvious: They think Hamas will win."
It is not clear when local elections will be held in these towns. Some PA officials have suggested that the third and last phase of the elections might be held before the year's end.
Khalid Amayreh in the West Bank contributed to this report