The 28-year-old England international was sacked by Sunderland last month after pleading guilty to two child sex offences. He was convicted last week of sexual activity with a child and now faces up to 10 years in prison.
"The Board of Sunderland AFC has today accepted the resignation of Margaret Byrne," the club said in a statement.
"Margaret, in her role as CEO, was responsible for the running of the club. She was also accountable for the actions taken by the club in relation to Mr Johnson," Sunderland said.
"Sunderland AFC acknowledges that Margaret's intentions have always been to act in the best interests of the club, however it has become clear through our own internal investigations that in this instance decisions have been taken by Margaret in error."
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Johnson was suspended by Sunderland in March 2015 after being arrested on suspicion of sexual activity with the 15-year-old. But two weeks later, after his bail was extended, the club lifted the ban.
He played 28 games for the team before being sacked just before his trial in February, when he pleaded guilty to grooming a girl and a charge of sexual activity.
"Contrary to what has been suggested, I did not understand that Mr Johnson intended to change his plea at trial or at all," Byrne said in a statement. "I was astounded when he did plead guilty."
But Byrne added: "I accept that Mr Johnson should not have been permitted to play again, irrespective of what he was going to plead. It was a serious error of judgement and I accept full responsibility for this."
Byrne, 35, joined Sunderland as club secretary in 2007 and became Chief Executive in 2011.
(Reporting by Ed Osmond; Editing by Kevin Liffey)
