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's tweet excluded mention of Former Prime Minister Nouri al-Maliki's State of Law Coalition, al-Fatih led by Shia militia leader Hadi al-Amiri, and the Patriotic Union of Kurdistan (PUK), all known for having close ties to Iran.
The race to become Iraq's next prime minister appears wide open as Muqtada al Sadr's alliance looked to be in the lead after the first elections since the defeat of Daesh. Mr. Sadr, who once called for attacks on American forces, capitalized on this widespread discontent by rebranding himself in recent years as a champion of the poor, a firebrand against corruption and a patriot who rails against outside interference by Iran as well as America.
Saturday's election is the first since the defeat of ISIL previous year.
Voter turnout was at a low 44 percent, 15 percent lower than the turnout in the 2014 parliamentary elections. It says it will announce the remaining results Tuesday.
Members of the national election commission read out vote tallies for each candidate list in each of the 10 provinces on national TV.
"If we want to change things then the prime minister needs to come from Marching Towards Reform", said resident Salah Jamal, 24.
The remaining uncounted ballots, mostly from Iraqis overseas, the security services, and internally displaced people voting in camps and elsewhere, might change the final seat tallies but only marginally. The other winning blocks, though, will have to approve his nomination.
A similar fate could befall Sadr.
Political sources told AFP that two meetings have been held under Iranian guidance to bring together several political blocs.
"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.