Sinn Féin deputy leader Michelle O'Neill has said the next few days are “crucial” for the return of a devolved government in Northern Ireland.
Its chairman, Sir Geoffrey Donaldson, said it provided a basis for a return to Stormont, subject to laws passed by Parliament.
Ms O'Neill said it was a “day of hope”.
Sinn Féin is the largest party in Northern Ireland after the last general election and will nominate Ms O'Neill for the post of first minister, while the DUP is the second largest and is eligible for the role of deputy first minister.
This is the first time that the Northern Ireland Assembly has had a Nationalist First Minister.
The DUP has blocked the executive and assembly from February 2022 in protest over post-Brexit trade arrangements.
Unionists said the checks on goods entering Northern Ireland from the rest of the United Kingdom weakened Northern Ireland's place in the union.
Northern Ireland Secretary Chris Heaton-Harris said on Wednesday he would release details of the deal between the government and the DUP, which would “protect Northern Ireland's place in the UK internal market”.
Mr Heaton-Harris will brief the parties on the content of the agreement at Stormont Castle on Tuesday afternoon.
Legislation is likely to be passed in Parliament on Thursday, leading to the recall of the assembly on Friday or Saturday.
The first order of business is choosing a new speaker – this should happen before any other business can be conducted.
The DUP has introduced additional checks on goods moving between Great Britain and Northern Ireland, which the DUP argued undermined Northern Ireland's position in the UK internal market.
Sir Geoffrey said the new law would “remove checks on goods left in the UK and in Northern Ireland, ending Northern Ireland's blind adherence to EU laws”.
He added: “There will be legislation protecting Union laws which guarantee unfettered access for Northern Ireland business to the rest of the UK.”
Whoever forms the next UK government, these measures will be taken as “legislative responsibilities”, he said.
He said his party's deal was not perfect but represented a “good decision” for Northern Ireland.
He hopes the government can move “quickly” to introduce legislation to implement what was agreed.
Speaking at a press conference on Tuesday afternoon, Sir Geoffrey said it was “remarkable that agreement has been reached, including when measured against the DUP's seven tests”.
He said “there will no longer be physical checks or identity checks on goods moving between Great Britain and Northern Ireland, as usual anywhere in the UK, where there is suspicion of criminal smuggling”.
“Once again in customs documents, customs declarations, supplementary declarations will disappear, which we believe represents a significant change.”
Despite the remote location of Monday's DUP meeting being kept secret, senior party members were confronted by protesters carrying “sell-out” banners.
A DUP executive wearing a wire was behind the leak of the meeting, which broadcast the party leader's speech to loyalist activist Jamie Bryson, BBC News NI understands.
“Obviously I wasn't in the room, and that place was clearly not survived by some hidden device, so clearly the seniors within the DUP – the plural – felt very strongly about it, they took this extraordinary, unprecedented step,” Mr. Bryson explained.
Has the target been achieved?
Sir Geoffrey Donaldson, I think, is quite happy to go beyond this.
Many doubted whether this would be possible.
I have to say that last night we were very much in that camp as the meeting lasted for five hours.
There was a feeling outside the meeting that things were not going well for the party leader.
So there was some surprise in the chamber when he confirmed that his party was back on the path to devolution, on the condition that the law would come into force earlier.
Sir Geoffrey looked very relieved because the meeting had been bruising.
There are still divisions within the party, but I think he very much felt that it had been accomplished.
But Sir Geoffrey's challenge now is to deal with the fallout.
Speaking in Stormont's Great Hall on Tuesday, Sinn Féin leader Mary Lou McDonald said she was pleased Northern Ireland was “on the cutting edge of government reform”.
He added that Mrs O'Neill's inauguration as Northern Ireland's first nationalist first minister was a “huge moment”.
He said it was “a scale of change across the north and right of Ireland”, but admitted the “sequence” of a return to devolution still needed to be agreed.
Coalition leader Naomi Long said the DUP's deal brought “bitter feelings” because of the time away from Stormont.
Ulster Unionist Party leader Doug Beattie said the deal appeared to meet only three of the DUP's seven tests.
Social Democratic and Labor Party (SDLP) lawmaker Matthew O'Toole described Monday night's events as “very surreal” but hoped it would lead to progress towards devolution.
Traditional Unionist Voice leader Jim Allister said there was a “need to separate fact from fiction” and accused the DUP of a “monstrous hike” in the deal with the government.
Prime Minister Rishi Sunak and Taoiseach (Irish Prime Minister) Leo Varadkar both welcomed the developments.
Both leaders hope this will now pave the way for the restoration of Northern Ireland's executive and legislature, and the resumption of north-south ministerial meetings.
Ireland's Minister for Industry, Trade and Employment, Simon Coveney, said the Irish government had not seen a draft of the deal agreed between the DUP and the UK government.
However, he said his government was committed to ensuring that any new arrangements that undermined the Windsor framework would not undermine trade agreements between Ireland and the UK that facilitate trade. EU and UK”.
Sir Geoffrey said he had given his word to unions that public sector workers would be rewarded when companies were restored.
Two years without devolution have increased budget pressures. Budgets are set at Westminster and no major policy or spending decisions can be made in Northern Ireland without ministers.
More than 100,000 public sector workers, including nurses, teachers, transport workers and civil servants, staged a mass walkout demanding pay a fortnight ago.