According to the officials' latest announcement, the political coalition of Shia Muslim cleric Muqtada al-Sadr has an early lead.
Official results are expected to be announced by the end of Monday, 48 hours after polling stations around the country closed.
The Shia-led administration of Mr Abadi has won praise because of the struggle against IS militants, also security has vastly improved throughout the country. Sadr has kept Tehran at a distance.
Abadi, a Shi'a who has sought to balance the competing influence of Washington and Tehran, finished third in six provinces but came in fifth in the capital.
Unlike previous elections, new electronic devices were used in Saturday's parliamentary elections to check election cards, fingerprints, and count ballots-a system meant to streamline the process and prevent fraud from taking place.
Iranian-backed Shi'ite militia commander Hadi al-Amiri's Fatah Alliance closely followed in second place, while Abadi's Victory Alliance appeared to be performing poorly across majority Shi'ite provinces that normally would be a strong base of support.
Moktada al-Sadr Leads In Iraqi Election Count
The tallied votes included full returns from only 10 of the country's 19 provinces.
Many Iraqis fear that their country can once again turn into a casualty in virtually any conflict involving America and Iran, our correspondent adds.
Sadr derives much of his authority from his family.
Murdered Sadr has formed an unlikely alliance with communists and other independent secular supporters who joined protests he organised in 2016 to press the government to see through a move to stem endemic corruption.
Whoever wins the election will have to contend with the fallout from USA president Donald Trump's decision to quit Iran's nuclear deal, a move Iraqis fear could turn their country into a theatre of conflict between Washington and Tehran. Al-Sadr commanded fighters in the war against the IS group and headed a powerful militia that fought USA forces in Iraq prior to 2011, but his 2018 campaign focused on social issues and eliminating government corruption.
Sadr portrays himself as an Iraqi nationalist and a year ago met Crown Prince Mohammed bin Salman of Saudi Arabia who is staunchly opposed to Iran.
The cross-border attack by the Iran-allied rebels, known as Houthis, came amid mounting regional tensions after U.S. President Donald Trump chose to withdraw from the landmark 2015 nuclear agreement with Tehran.
The last time the Warriors lost two consecutive playoff games was the 2016 NBA Finals when they lost Games 3 and 4 at Cleveland. Anthony Davis proved to be a beneficiary of Rondo's assists as he scored 14 of his 33 total points in the third quarter alone.
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