Former Prime Minister Nuri al-Maliki, a close ally of Iran like Amiri, came in fourth with around 25 seats.
Moqtada al-Sadr, the Shia cleric whose Mahdi Army bloodied the nose of the USA military in the battle of Fallujah in 2004 that left eight Kellogg, Brown and Root transport drivers and three US soldiers dead and who was considered an "outlaw", has won Iraq's parliamentary election.
Al-Sadr said in Tweet he was open to forming a coalition with al-Abadi to form a new government for Iraq.
He can not become prime minister as he did not run in the election, though his apparent victory puts him in a position to pick someone for the job.
The populist Shia cleric on Tuesday dealt a blow to both Iranian and American influence with a shock election triumph that has upended Iraqi politics.
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Sadr has led two uprisings against U.S. forces in Iraq and is one of the few Shia leaders to distance himself from Iran.
Victory for the veteran nationalist's Marching Towards Reform alliance with Iraq's communists - pitched as an anti-corruption outsider force - would be a slap in the face for Iraq's widely reviled ruling establishment.
No single group is expected to gain an outright majority.
This would be a blow for the US-backed Prime Minister Haider al-Abadi's al-Nasr bloc, along with former premier Nouri al-Maliki's State of Law alliance and the Iranian-backed, semi-political wing of the Hashd al-Shaabi militias called al-Fatah.
The elections on Saturday - hit by record abstentions - saw a clear rejection of the Iraqi elite that has run the country since the ouster of Saddam Hussein in 2003. Despite a third place finish, Abadi could potentially still remain prime minister after the government coalition is formed.
Iraqi firebrand political figure Muqtada al-Sadr is set to be announced the surprise victor of the country's elections and prepared for his new status as government titan by making a call for national unity.