The vote is the first being held since the defeat of Islamic State (ISIS) terrorists in the country.
The ballot saw a record low turnout, as only 44.5 percent of eligible voters headed to the polls in the lowest participation rate since the 2003 US-led ouster of Saddam Hussein.
Results from eight provinces are yet to be announced, including from Nineveh - which has the second-largest number of seats after the capital Baghdad.
An official at the US State Department remained coy ahead of the definitive tally, telling Agence France-Presse "we are awaiting the announcement of the official results and look forward to the formation of the new government".
According to Rudaw, "Sadr's apparent victory in Baghdad could have an impact as Shiite-dominated Iraq seeks to form a coalition government after Ramadan", which begins on Tuesday. It is also possible for al-Sadr and al-Abadi to join forces which could result in al-Abadi being named prime minister again.
Supporters of Shiite cleric Moqtada al-Sadr carry his image as they celebrate in Tahrir Square, Baghdad, on May 14.
Abadi - who came to power as IS swept across Iraq in 2014 - has been a consensus figure who balanced off the United States and Iran.
Forty-four year old Sadr will not become prime minister as he did not run in the election but his nearly certain victory puts him in a position to pick someone for the job.
Sadr has led two uprisings against USA forces in Iraq and is one of the few Shia leaders to distance himself from Iran. Authorities are seeking as much as $88 billion for postwar reconstruction. His father's cousin, Mohammed Baqir, was killed by Saddam in 1980.
A document being circulated among journalists and analysts by a candidate in Baghdad showed Sadr had won the nationwide popular vote with over 1.3 million votes, followed by Amiri with over 1.2 million and Abadi with over 1 million.
Although he is primed to receive a considerable amount of votes, he does not have Tehran's backing and thought to be failed to muster an overall majority without striking a tricky coalition agreement with other blocs.
In a further political upset, the rival Conquest Alliance of pro-Iranian former fighters appeared to be coming in second, squeezing internationally favored prime minister Haider al-Abadi into 3rd.
Political power in Iraq is traditionally divided along sectarian lines among the offices of prime minister, president and parliament speaker.
Abadi was seen by some Iraqis as lacking charisma and ineffective.
Alternatively, these blocs could sideline Abadi altogether if they can muster a coalition of their own with the formula and size to secure power.
During the campaign, frustrated Iraqis of all shades complained about their political elite's systematic patronage, bad governance and corruption, saying they did not receive any benefits of their country's oil wealth.
Amiri's Badr organisation played a key role in the battle against Islamic State.
He invited a US-led coalition to help roll back the jihadists and kept up good ties with Tehran. The dissident-turned-militia leader spent more than two decades fighting Saddam from exile in Iran. Winning the largest number of seats does not automatically guarantee that, however.