In the first coronial inquest his death was ruled to be a suicide, despite his clothes being discovered neatly laid out at the top of the cliff above.
However, the third inquest in 2017 found Mr Johnson fell from the clifftop due to violence by an unidentified attacker who perceived him to be gay.
During a pre-trial hearing in Sydney on Monday, White, who had previously pleaded not guilty to murder, plead guilty.
On Thursday, a New South Wales Supreme Court judge accepted the guilty plea.
Mr Johnson's brother Steve Johnson, who travelled to Australia to attend the pre-trial hearing this week, said on Friday the guilty plea triggered “a mixture of emotions”, with relief chief among them.
“This is not what any of us expected this week,” he told the ABC.
“We all sat down on Monday and the judge read how the defendant pleaded and he surprised everybody in the courtroom by saying 'guilty'. It turned everything upside down. We weren't expecting it.
“The police were sure they had the right person but you're never sure until you hear those words from the prisoner himself – and so suddenly, I knew who killed my brother.”
A 2018 police review of 88 suspicious deaths in NSW between 1976 and 2000 revealed 27 men were likely murdered due to their homosexuality, with cases peaking in the late 1980s and early 1990s.
White, 49, is in custody and will be sentenced on 2 May.
“I actually appreciate that Scott White founded in his role to spare us more months of this gruelling journey, and a trial that's scheduled for May it no longer is necessary,” Steve Johnson said.
“I am hopeful for the other dozens of gay men who lost their lives around the same time that Scott did.”
Steve Johnson said he never felt like he “had any other choice” other than to keep “pushing for justice for Scott” over the last 33 years.
“I was sure from the beginning this couldn't have been a suicide,” he said.
Steve Johnson said he was “very impressed with the way that the justice system worked”.
“I certainly had a lot of time to stare at Scott White and imagine what might have happened at the cliffs. The court gave permission to police to take me through the evidence today and that's something I'll be doing today before I leave Sydney, so I'll learn a lot more about the last few hours of my brother's life.”