Speaking to BBC Radio Ulster's Good Morning Ulster programme Brian Stack's son, Austin Stack, said the apology "means an awful lot to us".
"It's a great sense of comfort, to particularly my mother and to myself and my two brothers, because the state is officially owning up to the errors and the missions that happened," he said.
He went on to say that the serious crime review team looked into the first investigation and "they found 194 errors and omissions".
"So I have, and my family have, met with several ministers for justice down through the years and we have a requested sight of that report.
"And every time we've requested it, we've been told that it's an internally valid document and that we cannot see it."
Austin Stack said that Wednesday would be a very emotional day for his family.
"My dad was doing a very difficult job at a very difficult time in our history," he said.
"He was awarded the Irish Prison Service's highest bravery award for that work that he was doing. And this is further vindication of him and his good name."
Prior to the Dáil proceedings, the Stack family will hold a private meeting at government buildings with the taoiseach and Minister for Justice Jim O'Callaghan.
The IRA admitted killing Stack in 2013 after his family mounted a long campaign to secure the admission, and met Sinn Fein's Gerry Adams and a former IRA commander.
Adams, at that time, expressed his "regret" to the Stack family over the killing.
A Sinn Féin spokesperson said that "in a 2013 statement, a former IRA leader acknowledged that IRA volunteers were responsible for Brian Stack's killing".
"The statement said the shooting had not been authorised by the IRA leadership, accepted IRA responsibility for his death, said it should not have happened and expressed sorrow for the pain and hurt suffered by the family."
The spokesperson added: "As republicans, we acknowledge the hurt caused by republican actions. That suffering cannot be disowned. We are sorry for all the lives lost during the period of conflict on this island, without exception."
In 2019, the family received an apology from then garda commissioner Drew Harris in relation to the handling of the investigation into the murder.
In 2023, gardaí told the family that the file on the case - which was with the DPP (Director of Public Prosecutions) - was being sent back to police for further investigation.
Stack was the only prisoner officer in the state killed by violence related to the Troubles in Northern Ireland.