The vote underpins the Hillsborough Agreement brokered between the Democratic Unionist Party (DUP) and Sinn Fein to stabilise the region's power-sharing government.
The Assembly decision followed an acrimonious debate in which the Ulster Unionists resisted pressure to support the move, which will now lead to the creation of a Department of Justice for Northern Ireland after the powers are devolved by April 12.
Out of the 105 votes cast, a total of 88 were in support of the move, with only the UUP voting against.
Democratic Unionist leader and Northern Ireland First Minister Peter Robinson attacked the UUP decision to vote against the devolution plans.
He said the party was seeking political advantage and added: "I believe it is time for us all to move forward. There must be no going back to the bad old days of the past.
"Throughout history there are times of challenge and defining moments. This is such a time. This is such a moment."
Mr Robinson added: "Leadership is not about what's easiest, or what best suits our party interests, it is about doing what is right for our people."
The UUP claimed the ministerial Executive must sort out other outstanding matters - such as the longstanding row over post-primary school transfer arrangements - before it can take on law and order functions.
Its leader Sir Reg Empey said: "I am immensely proud of the sacrifices my party has made for the cause of peace. Our determination to make Stormont work for all the people of Northern Ireland - unionists, nationalists, all of us - continues.
"Our whole-hearted support for the brave men and women of the PSNI continues unabated."
Sir Reg added: "As a democratic political party pledged to making power-sharing work in an inclusive manner for all the people of Northern Ireland - and we exercise our rights refusing to bow to the blackmail and bullying to which we have been subjected in recent weeks."
Deputy First Minister Martin McGuinness questioned the logic of the UUP stance.
"The UUP are on record as stating that they will not support the election of a local minister to oversee the administration of policing and justice until there is agreement to test 10 and 11-year-old children to determine which school they will attend," he said.
"I have to say that these are the most dysfunctional political positions I have ever come across."
He added: "The Hillsborough Agreement provided an opportunity for a new start... I acknowledge that this is a challenge to us all but it is one to which we all must rise."
Prime Minister Gordon Brown said: "Today the politics of progress have finally replaced the politics of division in Northern Ireland.
"The completion of devolution, supported by all sections of the community in Northern Ireland, is the final end to decades of strife.
"It sends the most powerful message to those who would return to violence: that democracy and tolerance will prevail.
"The courage and leadership of the parties who voted to complete devolution at Stormont will be noted around the world."
The Hillsborough Castle deal, signed after nearly two weeks of round-the-clock talks at the Co Down venue last month, promised delivery of the republican demand for the devolution of policing and justice powers, plus the unionist call for the creation of new systems to oversee loyal order parades.
The agreement was aimed at providing greater stability to the powersharing administration, avoiding a threatened collapse of the institutions after a lengthy political stand-off on the devolution issue.
The UUP was accused of electioneering over the deal in a bid to put its unionist rivals, the DUP, under pressure, though the party strenuously denied the allegation.
Hours before the vote, the widow of a police officer murdered by dissident republicans made a dramatic plea for all politicians to back the deal.
Kate Carroll, whose husband Pc Stephen Carroll was shot dead by the Continuity IRA a year ago today, said all parties should support the plan.
In a surprise call to a radio show, she said: "This morning has been very, very hard for me, and I would just ask everyone in Stormont to please get on with their job."
Mrs Carroll told the UUP: "I am pleading on this day that is so important to me that it's not worth it. Life is too short. It is heartbreaking that I have to get on this morning to please ask the politicians to get on with their job."
The DUP and Sinn Fein had the strength in numbers to pass today's measures, but the rejection from the UUP deprived them of the unanimous support they and the British and Irish Governments had sought for the devolution deal.
Sinn Fein's 28 Assembly members and the nationalist SDLP's 16 backed the devolution plan. The Alliance Party's seven members, together with the single Green party member and an independent member also backed the move.
On the unionist benches, 34 of the DUP's 36-strong team voted "Yes", with the party's Willie McCrea said to be absent because he was attending a funeral, while the party's Willie Hay serves as Assembly Speaker.
The Progressive Unionist Party's leader Dawn Purvis also voted in support of the devolution plan.
The UUP was without one of its members who was absent for personal reasons, but the remaining 17 Ulster Unionists voted against the motion.
Over recent days the UUP came under international pressure to back the deal, with Conservative leader David Cameron confirming he was asked to intervene in the issue by former US President George Bush.
Mr Cameron, whose party is in an electoral pact with the UUP, said Conservatives had played the "most constructive role we possibly could as an Opposition" in relation to Northern Ireland.
But he added: "The one thing we cannot do is force people to vote a particular way."
Leader of the nationalist SDLP Margaret Ritchie said her party would vote in support of the devolution of policing and justice powers, but she was highly critical of the wider Hillsborough deal.
She condemned plans to step outside the normal powersharing selection process for ministerial posts, with the move set to see the smaller Alliance Party emerge as an agreed candidate for the new post of Justice Minister.
Alliance Party leader David Ford, who is the favourite to be Justice Minister when the powers are devolved, said the move would cement the peace process.
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