The United Nations had criticised the change as unfair to Iraq's Sunni Arab minority, which had threatened to boycott the vote.
After a brief debate on Wednesday, the Assembly voted 119 to 28 to restore the original voting rules for the referendum, which will take place 15 October. Only about half of the 275-member legislative body turned up for the vote.
Washington hopes a majority "yes" vote in the referendum will unite Iraq's disparate factions and erode support for fighters in Iraq, paving the way to eventually begin withdrawing foreign troops.
US and UN officials hope that restoring the original rules will avert a boycott of the referendum by the Sunni minority, which would undermine the credibility of the vote and wreck efforts to bring Sunnis into the political process.
Many Sunnis oppose the charter and want it rewritten, believing it would divide Iraq and leave the Shia in the south and Kurds in the north with virtual autonomy and control over Iraq's oil wealth, while isolating Sunnis with little power or revenue in central and western areas.
Original rules restored
The original rules, now restored, mean that Sunnis can veto the constitution by getting a two-thirds "no" vote in three provinces, even if the charter wins majority approval nationwide. Sunnis have a sufficient majority in four of Iraq's 18 provinces.
"Ultimately, this will be a sovereign decision by the Iraqis, and it's up to the Iraqi National Assembly to decide on the appropriate electoral framework"
UN spokesman in New York
On Sunday, Iraq's Shia- and Kurdish-controlled parliament effectively closed that loophole with their rule change. The legislature decided that a simple majority of those who cast votes means the constitution's victory - but that two-thirds of registered voters must cast "no" ballots in three provinces to defeat it.
That interpretation had raised the bar to a level almost impossible to meet. In a province of 1 million registered voters, for example, 660,000 would have to vote "no" - even if that many didn't go to the polls.
US, UN unhappy
In behind-the-scenes negotiations on Tuesday, US and UN officials pressed Iraqi legislators and government officials to reverse that change.
The UN said the change was a violation of international standards.
"Ultimately, this will be a sovereign decision by the Iraqis, and it's up to the Iraqi National Assembly to decide on the appropriate electoral framework," UN spokesman Stephane Dujarric said in New York on Tuesday.
"That being said, it is our duty in our role in Iraq to point out when the process does not meet international standards."
The Americans were talking separately with the Shia-led government.
Distribution of constitution
On Monday, UN officials began distributing 5 million copies of the constitution to voters across Iraq. The world body also will monitor the voting on 15 October.
The UN criticised the change
After Wednesday's vote, the deputy speaker, Hussein al-Shahristani, said the parliament now agreed that the word "voter" throughout the election rules means someone "who did really cast his vote in the referendum" - both for the purposes of passing the referendum or for getting the two-thirds threshold needed to defeat.
"The government is completely keen to make the constitutional process legitimate and of high credibility and we are concerned about the success of this process rather than the results of the referendum," government spokesman Laith Kubba said after the vote.