Tallies put the anti-establishment Conquest Alliance of pro-Iranian former paramilitary fighters who helped battle ISIS in second, and incumbent Prime Minister Haider al-Abadi's bloc back in third.
Sadr faces a hard act to herd together enough groups from across Iraq's fragmented political spectrum to form a government.
Strengthened by his apparent victory in Saturday's polls, black-turbaned Sadr could now push his nationalist agenda that has seen him pledge to curb foreign meddling in Iraq.
While speculation swirls, the next concrete step remains completing the vote count and firming up the final makeup of Iraq's new 329-seat parliament.
Saturday's election is the first since the defeat of ISIL past year.
Abadi - who came to power in 2014 as IS rampaged across Iraq - has balanced off the United States and Iran during his time at the helm.
Mariners' Cano gets 80-game ban, loses more than $11M in salary
Whether Cano reaches that mark or not, his Hall of Fame case will include questions about the legitimacy of his accomplishments. A source told MLB.com that Cano has been tested at least once since the positive test result.
Turnout was 44.52 percent with 92 percent of votes counted, the Independent High Electoral Commission said, the lowest participation rate in Iraq's post-Saddam history. It says it will announce the remaining results Tuesday. It included full returns from only 10 of the country's 19 provinces, including the provinces of Baghdad and Basra.
The elections commission has announced results from 16 of Iraq's 19 provinces.
Also in the mix is ex-prime minister Nuri al Maliki, a divisive figure blamed for losing territory to Daesh and stirring sectarianism, who has cultivated ties with Iran. Of more than 2 million Iraqis displaced by the war, the majority are Sunnis.
At elections in 2010, the Iraqi National Movement of Ayad Allawi - loathed by Iran - scooped 91 seats to become the biggest group in parliament.
A similar fate could befall Sadr.
"We will not allow liberals and communists to govern in Iraq", Ali Akbar Velayati, top adviser to the Islamic Republic's Supreme Leader Ayatollah Ali Khamenei, said in February.