The assembly passed a motion calling on the Irish Government to apologise for its alleged role in the emergence of the IRA in the Troubles, on 17 September 2012.
The DUP's Gregory Campbell presented his party's motion and said the issue of dealing with the past was "fraught with difficulties".
Mr Campbell said the motion did not seek an apology from the Irish government or the Irish prime minister for the actions of the IRA.
"What we do seek is the predecessor of Enda Kenny, Jack Lynch, in government as Taoiseach of the Irish Republic, with the full knowledge of members of his cabinet, he allowed money to be given to the emerging IRA, and it's for that we ask him (Mr Kenny) to apologise," he said.
His party colleague, Arlene Foster, also said the apology "must have reference to and acknowledge the ethnic cleansing" which, she said, took place along the border by the provisional IRA.
"The story of the Fermanagh border in the 70s, 80s and 90s was that nobody was there, security was not an issue that the Republic's government got involved with when Protestant only sons were being ethnically cleansed from the area," she said.
Sinn Fein's Mitchel McLaughlin said unionists needed to "get real" over the history of Northern Ireland and the role of the old Stormont.
He suggested the DUP and others were in denial about the past and he accused Mr Campbell of being selective.
"I don't know know any republican who would say that there wasn't fault on all sides," he said
Ulster Unionist Danny Kennedy presented his party's amendment which noted "the apologies from the United Kingdom government in relation to the Bloody Sunday and Claudy bombing".
He said he supported the DUP motion and he had already asked the Irish Prime Minister Enda Kenny to apologise on behalf of his people "for the security failings and the political failings of successive Irish governments, failings which may not have caused the atrocities such as Kingsmills, but failings which meant that little or nothing was done".
Last week, Mr Kenny told relatives of the ten textile workers killed in the 1976 Kingsmills massacre in County Armagh, he could not apologise for the IRA.
Stewart Dickson spoke on behalf of the Alliance Party's amendment calling for Secretary of State Theresa Villiers and her counterparts in the Irish government to convene talks between all political parties on the subject of dealing with the past.
"We should establish a process to allow us to examine very deep, hurtful and painful issues," he said.
Conall McDevitt of the SDLP described the motion as "premature" and said that the motion was "not the way" to deal with the past.
TUV leader Jim Allister said "an apology could do with starting nearer to home".
The UUP amendment passed, and the Alliance Party's fell.
Despite opposition from the SDLP and Sinn Fein, the amended motion passed by 47 votes to 46.