Shi'ite militia leader Hadi al-Amiri's bloc, which is backed by Tehran, is in second place.
The vote is the first being held since the defeat of Islamic State (ISIS) terrorists in the country.
The electoral commission of Iraq announced that 44.5 percent of those eligible had cast their ballots in the elections. If confirmed once the final election results are announced, this percentage will be overshadowed by Lebanon's parliamentary election on May 6, which saw 49.2 percent of the electorate head for the polls, and will trump the Egyptian presidential election in March of this year, which saw a turnout of 41 percent.
Supporters of Shiite cleric Moqtada al-Sadr carry his image as they celebrate in Tahrir Square, Baghdad, on May 14. The National Wisdom Movement (al-Hikm) is led by Ammar al-Hakim, former leader of the Islamic Supreme Council of Iraq (ISCI).
Sadr has reinvented himself as an anti-graft crusader after rising to prominence as a powerful militia chief whose fighters battled USA troops after the 2003 invasion.
In the cleric's impoverished Sadr City neighbourhood in Baghdad residents said they were hoping for improvements after the elections.
But Sadr has grown increasingly pragmatic over the years and formed a cross-sectarian electoral alliance emphasizing Iraqi nationalism over loyalty to Iranian clerics and American military and political backing.
The winners not only scramble the pyramid of power in Iraq but also raise the possibility of a government with radically new priorities.
He was viewed as a frontrunner before the election.
Congratulating the nation and government of Iraq for the election, Iran hopes to further cement the friendly relations between the two countries, the statement highlighted. Sadr refused to adhere to the calls from Tehran and Qom to send troops to support the Assad regime in Syria, and tensions have remained high.
Abadi was seen by some Iraqis as lacking charisma and ineffective. He had no powerful political machine of his own when he took office.
Amiri's Badr organisation played a key role in the battle against Islamic State.
Sadr ran a campaign of Arab nationalism, uniting Shiites but rejecting the intervention of Persian Shiites in Iraq political affairs.
Al-Amiri's Fatah Coalition, now in second place, has close ties to Iran.
The Victory Alliance of Prime Minister Haider al-Abadi, who has been backed by the global community, looked to have won in only one province.