Sarah McGuire, 38, who was only 10-years-old at the time, told of the discovery - a week after Elaine's naked body had been found near her Greenock home in June 1986.
She recalled the incident at the High Court in Edinburgh yesterday, where John Docherty, 49, is on trial for the murder.
Ms McGuire told how she and a friend regularly visited the James Watt Library in Greenock's Union Street in the evening.
They cycled there together on June 9, 1986.
They took their bikes up the steps to the side of the front door and dumped them, leaving the library again about 20 or 30 minutes later.
Ms McGuire said: "I noticed a handbag.
"I remember it being navy blue. It was what you would refer to as a clutch bag.
"It didn't have a strap.
"It wasn't very big and it was burned, partially.
"There was a substantial part of the blue bag left but a section of it had been burned."
She said they told her friend's mum what they had seen and police took away the handbag.
Ms McGuire told the trial she did not remember seeing the bag on her way into the library.
Another witness, Michelle Burns, 46, said she had been to a Sunday night disco at the Celtic Club in Greenock's Laird Street on June 1, 1986.
Elaine Doyle was also dancing there that night, the trial has heard.
Ms Burns said she knew Elaine because the teenager sometimes came into the fish and chip shop where she worked.
She told how she saw Elaine later that night, alone and walking in the direction of her home.
Elaine was carrying a clutch bag said Ms Burns.
She told police later: "When I spoke to her she was OK, quite happy, her normal self."
In a statement 10 years later, she confirmed: "I don't remember seeing anything out of the ordinary."
Yesterday, Ms Burns told advocate depute John Scullion, prosecuting, that she did not see anyone with Elaine or anyone following her.
Docherty, now of Hunters Quay Holiday Village, Dunoon, denies murder and claims that at the time he is alleged to have stripped and strangled Elaine Doyle, 16, he was at home with his parents, who are no longer alive.
Docherty has also lodged a so-called special defence of incrimination claimed the culprit might be among a list of 41 names taken from files of the police investigation into the murder.
He also denies stealing a handbag from Ardgowan Street on the same date.
The trial continues.