The FA has come under fire for its selection of Sheffield Wednesday's ground to host the 1989 FA Cup semi-final between Liverpool and Nottingham Forest, even though it did not have a valid safety certificate.
English football's governing body has also been accused of not doing enough to explode the myth that Liverpool fans were responsible for the tragedy.
Wednesday's Hillsborough Independent Panel Report exonerated Liverpool supporters, and exposed the shocking extent of a police cover-up that attempted to absolve the force of blame.
Bernstein said in a statement: "We are deeply sorry this tragedy occurred at a venue the FA selected.
"The 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 all of the families of those who lost their lives and to everyone connected with the city of Liverpool and Liverpool Football Club.
"This should never have happened. Nobody should lose their lives when setting out to attend a football match and it is a matter of extreme regret and sadness that it has taken so long for these findings to be published and the truth to be told."
Earlier, Boris Johnson said he was "very, very sorry" for comments in a 2004 Spectator article about the Hillsborough disaster.
The London Mayor was editor of the Spectator magazine when its leading article criticised the city of Liverpool for its "failure to acknowledge ... the part played in the disaster by drunken fans".
Johnson, visited Merseyside to apologise shortly after the publication of the article, which also accused Liverpudlians of "mawkish sentimentality" in their reaction to the murder of civil engineer Ken Bigley in Iraq.
He said on Thursday: "I’m very, very glad that this (Hillsborough) report does lay to rest the false allegation that was made at the time about the behaviour of those fans.
"I was very, very sorry in 2004 that the Spectator did carry an editorial that partially repeated those allegations, I apologised then and I apologise now.
"I do hope the families of the 96 victims will take some comfort from this report and that they can reach some sort of closure."
A senior member of the police force who was involved in the safety operation at Hillsborough on the day of the disaster has announced he will not resign in spite of calls for him to do so.
Sir Norman Bettison, now chief constable of West Yorkshire Police, was a chief inspector at West Yorkshire Police in 1989. Following the release of Wednesday's report Trevor hicks, chairman of the Hillsborough Families Support Group, said Bettison should "scurry up a drainpipe" and resign for his perceived part in the tragedy.
However, Bettison released a lengthy statement on Thursday saying he had nothing to answer for.
The statement read in part: "In the absence of all the facts I was called upon to resign 14 years ago when I became chief constable of Merseyside," he said.
"I really welcome disclosure of all the facts that can be known about the Hillsborough tragedy because I have absolutely nothing to hide … Whilst not wishing to become a conducting rod for the genuine and justified hurt and anguish, I would invite anyone to do the same as me and read the document and the papers online."