The commission did not release any results from provinces where Sunni Muslim Arabs form the majority of the population. The results released so far on Monday show that the votes were divided along ethnic and sectarian lines.
The commission did not say how many people voted overall or provide further details.
In Baghdad province, results from 89% of the ballot boxes showed the Shia United Iraqi Alliance ahead with 58% of the vote in Iraq's biggest electoral district.
The electoral commission said the alliance received 1,403,901 votes, followed by the Sunni Arab Iraqi Accordance Front with 451,782 votes, and the Iraqi National List ticket of Iyad Allawi, the former prime minister, with 327,174 votes.
Baghdad is Iraq's biggest electoral district, with 2161 candidates running for 59 of parliament's 275 seats.
Baghdad is a mixed province but has a large Shia population.
Results from southern Basra province, also mixed but predominantly Shia, saw the clergy-backed alliance significantly ahead, winning 612,206 votes with 98% of ballot boxes counted. Allawi's group was in second place with 87,134 votes, while the Iraqi Accordance Front trailed with 36,997.
Kurdish parties were overwhelmingly ahead in their three northern provinces.
In Dahuk, results from 93% of ballot boxes showed the Kurdistan Coalition List, an alliance consisting of the two main Kurdish parties, received 344,717 votes representing 89% of votes counted.
Kurdish parties are ahead of
others in the northern provinces
The Kurdistan Islamic Union followed with 28,401 ballots, while the Rafidian party, which represents Assyrian Christians, trailed with 4696.
Allawi's group received 2327 votes.
In Arbil, results from 76% of ballot boxes showed the Kurdish alliance winning 570,098 votes, or 95%. The Kurdistan Islamic Union won 19,612, or 3.24%, while Allawi's ticket had 2420.
In Sulaimaniya, results from 98% of ballot boxes showed the Kurdish alliance ahead with 671,814 votes, followed by the Kurdistan Islamic Union with 83,208 - and trailed by Allawi's National List with 1806.
Demonstrations have broken out across Iraq in protest against the government's decision to raise the price of petrol, heating and cooking fuel anywhere from five- to nine-fold.
On Monday, Ibrahim Bahr al-Uloum, Iraq's oil minister, threatened to resign over the increase. A litre of petrol in Iraq still costs 17 cents or less, but Iraqis were upset by the size of the price increase.
Protesters burn tyres in Basra
over the increase in fuel prices
In Amarah, 290km southeast of Baghdad, police fired into the air to disperse the hundreds of protesters who had gathered in front of the provincial government headquarters. The demonstrators did not leave, and scuffles broke out with police.
Drivers blocked roads and set tyres on fire near fuel stations in the southern city of Basra, and hundreds demonstrated outside the governor's headquarters to protest against the increases.
The oil minister said when the cabinet increased prices, it decided that the additional money would be used to support more than two million low-income families so they would not be burdened by the increases.
The price of a litre of imported and super petrol was increased to 17 cents, but that represents a five-fold increase. The price of locally produced petrol was increased by about seven-fold to about 12 cents per litre.
There was no respite from violence on Monday.
Violence has reappeared after
a period of calm for the vote
A car bomb exploded outside a children's hospital in western Baghdad, killing at least two people and wounding 11, including seven police officers, officials said.
Police believe the bomb had been targeting a passing convoy carrying a police colonel, who was among the wounded.
In western Baghdad, armed men attacked the convoy of Ziad Tariq, the deputy governor, killing three civilians and wounding three of Tariq's bodyguards, Baghdad police said.
Tariq was not injured.