The child "sustained horrendous injuries", police said.
Despite the efforts of her family and paramedics, she died a short time later in Queen's Park in Bolton on Sunday afternoon.
The 30-year-old woman, who has "some history of mental illness", is not known to the girl's family.
She was detained at the scene by a member of the public, and arrested initially on suspicion of attempted murder and subsequently for murder.
Assistant Chief Constable Russ Jackson from the Greater Manchester force said "words cannot describe how awful this attack is".
He added: "The family were out in the park, enjoying the spring sun when in a totally unprovoked and random attack this little girl sustained horrendous injuries and sadly died a short while later.
"A 30-year-old woman who was not known to the family was detained by a member of the public and then arrested by the police.
"She is currently in custody for murder and it is expected she will be interviewed by detectives later."
He continued: "We are working to understand the motive for this completely random and brutal attack.
"We understand that the woman has some history of mental illness and we are working to understand if this played any part in her motive.
"We are determined to quickly understand how this came to happen, leaving a young family so distraught and so devastated in an instant."