Mr Corbyn got nearly 62% of the vote on Saturday, to beat his rival, Owen Smith, by a large margin.
More than half a million people voted, with Mr Corbyn receiving 313,209 of the votes cast, compared with 193,229 for Mr Smith.
Sinn Féin leader Gerry Adams described Mr Corbyn as a "stalwart friend to Ireland".
"I want to extend my warmest congratulations to Jeremy on his resounding win," Mr Adams said.
"He faces many challenges in the time ahead in confronting the right wing agenda of the Conservative government, not least in addressing the issue of Brexit.
"I wish him well and look forward to working with him in the time ahead to ensure that the gains of the peace process are built upon."
SDLP leader Mr Eastwood said: "This is a turbulent time in the politics of these islands as the Brexiteers seize control of the mechanics of government in Westminster.
"I look forward to working with the Labour leader as we continue to fight Brexit and promote the democratically stated will of people in Ireland."
The shadow NI secretary Dave Anderson told the BBC that he was "glad it is all over".
He said the leadership election was a distraction from holding the Conservative government to account.
Mr Anderson, who is a Corbyn supporter, said he hoped those MPs who voted for a motion of no confidence in the party leader would now have a change of heart.
The shadow NI secretary encouraged those critical of Mr Corbyn to return to their constituencies and "talk to your people".
The leadership announcement was made by Paddy Lillis from the ruling executive committee, who is a trade unionist from Northern Ireland.
After the result was announced, Mr Corbyn thanked voters for their "trust and support".
Labour Party activists from Northern Ireland are in Liverpool attending the conference. About a dozen party members and trade unionists from Northern Ireland have made the trip across the Irish Sea.
Kathryn Johnston, who is vice chair of Labour Party In Northern Ireland, watched the result being declared.
She told the BBC: "I am absolutely delighted. I did not start as a Jeremy Corbyn supporter, but I have been impressed with his commitment and direction."
She has long campaigned to persuade the party to stand candidates in Northern Ireland, but accepts that Mr Corbyn is not a supporter of this.
"He is not a fan of standing in Northern Ireland, it is up to us to persuade him," she said.
Ms Johnston and her colleagues are to hold a fringe meeting about organising in Northern Ireland.
Sinn Féin's Mary Lou Macdonald and Martin McGuinness are also attending the conference as will SDLP leader Mr Eastwood.