Johnson claims the 77 year old drugged her at his home in the 1980s during an audition for The Cosby Show.
In an article for Vanity Fair, she writes, "Cosby said he wanted to see how I handled various scenes, so he suggested that I pretend to be drunk...
"As I readied myself to be the best drunk I could be, he offered me a cappuccino from the espresso machine... He insisted that his espresso machine was the best model on the market and promised I'd never tasted a cappuccino quite like this one. It's nuts, I know, but it felt oddly inappropriate arguing with Bill Cosby so I took a few sips of the coffee just to appease him.
"Now let me explain this: I was a top model during the 70s, a period when drugs flowed at parties and photo shoots like bottled water at a health spa. I'd had my fun and experimented with my fair share of mood enhancers. I knew by the second sip of the drink Cosby had given me that I'd been drugged - and drugged good."
After realising that Cosby had allegedly slipped something into her drink, Johnson recalls launching into a tirade.
She continues, "As I felt my body go completely limp, my brain switched into automatic-survival mode. That meant making sure Cosby understood that I knew exactly what was happening at that very moment. (I said), 'You are a motherf**ker aren't you?'
"That's the exact question I yelled at him as he stood there holding me, expecting me to bend to his will. I rapidly called him several more 'motherf**kers'. By the fifth, I could tell that I was really p**sing him off. At one point he dropped his hands from my waist and just stood there looking at me like I'd lost my mind."
Upset Cosby subsequently kicked Johnson out of his home, she alleges, adding, "I recall his seething anger at my tirade and then him grabbing me by my left arm hard and yanking all 110 pounds of me down a bunch of stairs as my high heels clicked and clacked on every step. I feared my neck was going to break with the force he was using to pull me down those stairs.
"It was still late afternoon and the sun hadn't completely gone down yet. When we reached the front door, he pulled me outside of the brownstone and then, with his hand still tightly clenched around my arm, stood in the middle of the street waving down taxis.
"When one stopped, Cosby opened the door, shoved me into it and slammed the door behind me without ever saying a word."
Johnson attempted to call Cosby after the alleged incident, but his wife answered the phone.
Cosby is facing sexual abuse allegations from over 20 women, including other models and actresses, who claim they were drugged and assaulted during reported incidents which took place decades ago.
He has not addressed any of the claims against him but his lawyer and attorney insist all the allegations are false.
Two of his accusers have filed lawsuits in the past 10 days - one, Judy Huth, claims Cosby drugged and assaulted her at a Playboy Mansion party when she was 15.