In a statement the club said Johnson had "suffered unacceptably" while a youth team player.
Chelsea's apology comes after Mr Johnson, 57, said the London club had paid him £50,000 not to go public with allegations.
The club admitted that the confidentiality clause had been "inappropriate", especially in light of abuse on a wider scale in the game.
The club added that it now had "no desire to hide any historic abuse we uncover from view".
An external review would examine whether it carried out a proper investigation when the allegations first came to their attention in 2014.
Mr Johnson said he had been sexually abused by the club's former chief scout Eddie Heath, who died in the early 1980s.
Solicitor Tom West told Sky News he had been contacted by another former first team Chelsea player, who was remaining anonymous.
He too claimed to have been sexually assaulted by Heath.
Chelsea's statement came as at least 17 more former footballers came forward with allegations of sexual abuse within the game.
More than 20 players had already contacted the Professional Footballers' Association union to report historical sex abuse.
Some 18 police forces are now examining claims from more than 350 people, across 55 clubs.
Chelsea said they were "fully committed to ensuring the safety and well-being of all children and young people who are in our care or attending our premises".
The statement read: "Everyone at Chelsea Football Club has been profoundly shocked by news of historical child sex abuse across British football and our heart goes out to all the victims.
"We pay tribute to the enormous courage of the people who have spoken out about the horrific abuse which they endured, including former Chelsea player Gary Johnson.
"We recognise that to do so, after carrying the burden of those events for so long, must have been an extremely difficult thing to do.
"It is clear that Gary Johnson suffered unacceptably while in our employment in the 1970s for which the club apologises profusely.
"If further evidence is uncovered we will ensure that we help victims in any way we can.
"We certainly have no desire to hide any historic abuse we uncover from view. Quite the opposite.
"Chelsea today is a very different club than it was in the 1970s.
"We are fully committed to ensuring the safety and well-being of all children and young people who are in our care or attending our premises."