It is vying with the Democratic Unionist Party (DUP) – whose vote share dropped – for the entitlement to nominate the next first minister.
A unionist party has been the largest in government since Northern Ireland was formed in 1921.
The cross-community Alliance Party has made gains, with 79 of 90 seats filled.
......................................Sinn Féin winning the most seats would be a first for a party in Northern Ireland that designates as nationalist.
The DUP won 28 seats at the last assembly election in 2017, just ahead of Sinn Féin’s 27.
So far this time round, Sinn Féin has 23 Members of the Legislative Assembly (MLAs), the DUP has 22, Alliance has 16, the Ulster Unionist Party (UUP) has nine, and the Social Democratic and Labour Party (SDLP) has six after a poor performance.
Despite seeing a surge in vote share, the Traditional Unionist Voice (TUV) regained only one seat; one independent unionist has been elected and Gerry Carroll of People Before Profit was returned in Belfast West.
Counting is ongoing in three centres – Titanic Exhibition Centre in Belfast, Ulster University in Jordanstown and Meadowbank Sports Arena in Magherafelt – with the possibility that it could continue into Sunday.
Sinn Féin vice-president Michelle O’Neill said the election results represented a significant moment of change.
“It’s a defining moment for our politics and for our people,” she said after she was re-elected in Mid-Ulster.
“Today ushers in a new era which I believe presents us all with an opportunity to reimagine relationships in this society on the basis of fairness, on the basis of equality and on the basis of social justice.”
Ms O’Neill said she wanted to work through partnership, not division.
Earlier, DUP leader Sir Jeffrey Donaldson said his party would accept the outcome of the election, but that it would continue to press the government for changes to the Northern Ireland Protocol.
The protocol is an element of the UK government’s Brexit deal with the European Union which keeps Northern Ireland aligned with the EU’s single market for goods.
However, it has angered some unionists by placing additional trade barriers between Great Britain and Northern Ireland.
“We need to get the protocol issues sorted out, we need to see that Irish Sea border removed,” Sir Jeffrey told BBC News NI.
The ultimate goal of Sinn Féin is for Northern Ireland to leave the UK and become one country with the Republic of Ireland.
But a victory for Sinn Féin in this election does not mean a vote on Irish reunification – also known as a border poll – would be imminent.
The Northern Ireland Act 1998 – which followed the signing of the Good Friday Agreement – stated that Northern Ireland remained part of the United Kingdom and “shall not cease to be so without the consent of a majority of the people of Northern Ireland voting in a poll”
It also stated that the Northern Ireland Secretary would agree to hold a poll if it appeared likely that a majority of people wanted a united Ireland.
Successive opinion polls suggest that is not yet the case, with the most recent, published in April, putting support at about a third.
On Friday, Sinn Féin President Mary Lou McDonald said her party was on the cusp of securing the post of first minister and that planning for a unity referendum would come within a “five-year framework”.
While the office of the first and deputy first minister is an equal one with joint power, the allocation of the titles is regarded as symbolically important.
For Sinn Féin to be installed in the role, the majority of unionist assembly members would have to agree to power-sharing, as Northern Ireland operates under a system of mandatory coalition.
But DUP leader Sir Jeffrey Donaldson has said that he will not be nominating any ministers to the executive until the issue of the Northern Ireland Protocol is resolved.
The result of this election will have significance for the future of the Northern Ireland Protocol.
The assembly members who are elected will have to vote on whether to continue with the parts of the protocol which create the internal UK trade border.
That consent vote has to take place before the end of 2024. The vote will be decided by simple majority rather than requiring cross-community consent.
Unionist parties oppose the protocol whereas nationalists and the cross-community Alliance Party see it is an acceptable compromise to mitigate some of the impacts of Brexit.
The Northern Ireland Protocol has cast a long shadow over the election campaign following the resignation of First Minister Paul Givan in February.
The move by the DUP was in an effort to force the UK government to act over the post-Brexit trading arrangements.
Northern Ireland Secretary Brandon Lewis has indicated that the government will not be introducing legislation relating to the protocol in the Queen’s Speech next week.
