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.
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.
"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.
"The FA and English football has changed immeasurably, and has learnt many lessons in the last 23 years," Bernstein said.
