Deputy First Minister Michelle O'Neill has acknowledged Stormont's public health messaging was "undermined" by the Bobby Storey funeral controversy.
She said she regrets what happened after large numbers attended the veteran republican's funeral in June, and she wants to "rebuild trust" with the public.
Ms O'Neill was among the mourners, despite Covid-19 restrictions.
Other parties said their credibility was undermined.
It led to First Minister Arlene Foster stopping their joint news conferences but following Ms O'Neill's statement, this practice resumed on Thursday.
The two women shared a platform for the first time in more than two months, during which they outlined a new list of Covid-19 restrictions for specific areas with a high number of infections.
In her opening remarks, Ms O'Neill said: "I do accept that the public messaging about the pandemic has been undermined by the controversy over the last number of months.
"It was never my intention that that would happen, but it did and I regret that.
"So I want to rebuild trust with the public, as we have the most challenging time ahead of us."
'Hurt and anger'
The controversy led to a number of inquiries, including a police investigation into allegations that Covid-19 public health guidelines had been breached during the funeral.
Responding to Ms O'Neill's comments, Ms Foster said it was "right" that she had "acknowledged the undermining of the public messaging as a result of what happened, and the deep hurt and anger that was caused".
Ms Foster added that investigations into what happened at the funeral would "continue and we will have an outcome in relation to that".
However, she said people in Northern Ireland were now facing "a very serious period" in the pandemic and ministers need to deliver a message to help stop the spread of Covid-19.
During the press conference, Ms O'Neill was asked if the lack of joint press conferences had contributed to the current rise in infections, but she rejected that.
"The absence of a press conference didn't mean we weren't communicating a public message, we certainly were," she said.
"I'm glad that we're here today communicating it together."
Earlier, Taoiseach (Irish PM) Micheál Martin welcomed Ms O'Neill's statement of regret.
Mr Martin told RTÉ the funeral caused "damage to public confidence in the administration at the time", adding the response from Sinn Féin "for far too long was one of denial".
"It should have happened earlier - but it has happened and I welcome it," he added.
Analysis by BBC News NI political reporter Jayne McCormack
The timing of this admission by Michelle O'Neill made it all the more significant.
It came just a day after health officials delivered the starkest warning yet about the latest rise in Covid-19 cases.
Arlene Foster and Michelle O'Neill have not stood side-by-side at the press podiums for more than two months but this very much seemed directed at bringing an end to that stalemate.
It appears to have been enough to satisfy Mrs Foster's threshold to resume those briefings, after she said there had to be a "recognition" that the credibility of the executive's messaging had been breached.
There will still be those within the DUP who feel it's too little, too late but the party leadership has had the final say.
Speaking to RTÉ's Morning Ireland, Sinn Féin leader Mary Lou McDonald said she acknowledged the number of people at the funeral gave rise to public concerns.
She said: "Michelle, I think, is making clear an important point."Firstly her regret, and our regret, that any undermining would occur to the public health messaging.
"And also the fact that we need now, a joint effort across the island, and in the north between all of the parties in the executive, to stand on a common platform."
'Significant and helpful'
Earlier, DUP MP Gregory Campbell told BBC Radio Ulster's Talkback programme that it would be "symbolically important for the first and deputy first minister to speak with one voice".
However, he stressed there was potential for the resumption of a joint press conference to be "over-emphasised" and exaggerated, and that Sinn Féin had taken too long to "own up to their mistake".
His DUP colleague Carla Lockhart accused Michelle O'Neill of "breathtaking arrogance" on Twitter.
"The consequences of their blatant disregard for the public health message is evidenced in the case count, yet sorry seems to be the hardest word," said the Upper Bann MP.
The consequences of their blatant disregard for the public health message is evidenced in the case count, yet sorry seems to be the hardest word. Breathtaking arrogance. What they expect of others they won’t do themselves. https://t.co/VsLns2S5jT via @rte— Carla Lockhart MP (@carlalockhart) September 10, 2020
But Alliance MP for North Down, Stephen Farry, said Ms O'Neill's comments were "significant and helpful".
Ulster Unionist MLA Alan Chambers said there had "still been no proper apology" from Ms O'Neill.
Meanwhile, the DUP has criticised the Police Ombudsman for "a failure" to investigate how the PSNI handled the funeral.
Northern Ireland's Chief Constable Simon Byrne said police had forwarded a complaint from a member of the public regarding police actions leading up to the funeral on 30 June.
But the Police Ombudsman said it was not currently investigating the issue.
"We have not received any complaints from anyone with direct experience of the policing of a funeral in relation to Covid regulations, and are not currently investigating this issue," said a spokesperson.
DUP MLA and NI Policing Board member Joanne Bunting said it was "a weak defence for not acting".
Following a meeting with Police Ombudsman Marie Anderson on Tuesday, the East Belfast representative said: "The PSNI must be held fully accountable not just for proactive steps its officers take but for clear failures to act when public safety is at stake."
She said there was "deep unease" among many of those who elected her over "the Police Ombudsman's failure to provide scrutiny of PSNI inaction against large republican funerals which breached lockdown restrictions".
"This is in contrast to the commissioned PONI investigation into police practice which led to the issuing of fixed penalty notices at Black Lives Matter protests during the same period," she added.
"Such inconsistency does little to restore confidence in the role or added value of the Police Ombudsman's Office."
Following the funeral, Mr Byrne requested an external senior police officer oversee an investigation into potential breaches of Covid-19 restrictions.
Mark Webster, Cumbria Constabulary's deputy chief constable, was appointed to oversee and direct the PSNI's investigation into potential breaches of Covid-19 restrictions and also help review events leading up to the date of the funeral.
DCC Webster, who was nominated for the role by the National Police Chiefs' Council, said work on the investigation was "ongoing" at present.
"In due course I will make a further statement to provide an update on progress," he said.