The report by the Hillsborough Independent Panel, published on Wednesday, revealed a litany of police cover-ups in the wake of the horrific crush caused by overcrowding at the 1989 FA Cup semi-final between Liverpool and Nottingham Forest.
Prime Minister David Cameron made a public apology to the families of the victims on Wednesday, and there have already been calls for a fresh inquest and possible criminal charges brought against those responsible.
The hard-hitting report said more than 100 statements taken at the time had been doctored to remove evidence that portrayed the police in a negative light. It also disclosed a campaign by the authorities to blame Liverpool fans for the disaster.
Bernstein said the tragedy at Sheffield Wednesday's stadium should never have happened.
"Ninety-six people lost their lives at the FA Cup semi-final between Liverpool and Nottingham Forest 23 years ago," he said in a video statement on the FA's website on Thursday.
"We are deeply sorry this tragedy occurred at a venue the FA selected. This fixture was played in the FA's own competition, and on behalf of the Football Association I offer a full and unreserved apology and express sincere condolences to the families of all those who lost their lives and to everyone connected with the city of Liverpool and Liverpool FC.
"This should never have happened. Nobody should lose their life when setting out to attend a football match and it is a matter of extreme regret and with sadness that it has taken so long for these findings to be published and the truth to be told.
"For 23 years the families have suffered unbearable pain and we have profound sympathy with them."
The victims in the Hillsborough tragedy died in an overcrowded, fenced-in enclosure at the stadium's Leppings Lane end.
Harrowing images of fans trying to escape the crush, fans being carried away on makeshift stretchers made from ripped-up advertising boards, and bodies strewn around the pitch were beamed around the world.
The independent panel, set up in 2010 by the previous Labour government to examine all the public paperwork relating to the tragedy, concluded that it could have been avoided, more lives could have been saved, and that the police response was flawed.
"There were clear operational failures in response to the disaster and in its aftermath there were strenuous attempts to deflect the blame on to the fans," it said.
Hillsborough marked the lowest point in British football, which for decades leading up to the tragedy had been blighted by outdated facilities and hooliganism, ushering in a new era of modern family-friendly, all-seater stadiums.
"The FA and English football has changed immeasurably, and has learnt many lessons in the last 23 years," Bernstein said.
In an earlier statement the FA paid tribute to the families of the victims who have fought for more than two decades for justice for their loves ones.
"It is also important that the FA recognises the tireless commitment shown by so many, particularly the Hillsborough Family Support Group," it said.
"We welcome the publication of the report and the subsequent comments of the Prime Minister and the Leader of the Opposition."
Victims' families say they are to press for a new inquest into the disaster and criminal charges brought against those responsible.
"The truth is out today, the justice starts tomorrow," Trevor Hicks, the president of the Hillsborough Family Support Group and father of two teenage victims of the tragedy, said after the publication of the report.
"We will follow every avenue, from prosecution to changing the inquest verdicts. We are not looking for scapegoats, we are looking for accountability, and those responsible should hang their heads in shame."
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