He will be sentenced on 24 November.
The court heard Peacham stabbed Ms Rao in the head and crotch, then stole her handbag, mobile phone and laptop computer.
The 30-year-old was later found lying unconscious by a path in Castle Green Park with "catastrophic" injuries having been attacked "minutes" from her home.
"She is in a persistent vegetative state from which it is unlikely she will ever recover," prosecutor Julian Evans said.
Image copyright MET POLICE Image caption Barry Peacham was found guilty of wounding with intent and robbery/
Jurors heard Peacham had previous convictions dating back to 2008 including possessing an offensive weapon after he ran at people with a knife at a swimming pool.
He was also found guilty of another attack in Castle Green Park in 2010.
Following the verdict, Det Ch Insp Gary Holmes described Peacham as "depraved and vile" and "one of the most dangerous men I have ever dealt with".
"On that night he has effectively ended the life of a 30-year-old lady... in the most brutal and vicious manner, stabbing her in the head, penetrating her brain."