The paedophile DJ abused the children, and also one visitor, at Duncroft Approved School in Staines, Surrey.
He committed 46 offences, including rape and sexual assault, and had “unrestricted and largely unsupervised” access, said Surrey Police.
The abuse took place in locations including the kitchen, the principal’s office and a bedroom in the intensive care unit.
Savile also used his own vehicles, including a camper van, to abuse victims in school grounds.
The incidents took place during at least 16 visits between 1974 and 1979. Savile was twice allowed to stay at the school overnight.
The Crown Prosecution Service (CPS) said in December that no former staff would face charges following an investigation to discover if anyone was complicit in the abuse.
Surrey Police and prosecutors had the chance to bring Savile to justice in 2007 when they received four allegations about historical sexual offences, two of them at Duncroft in the 1970s.
The Surrey force consulted with the CPS but in October 2009 decided no prosecution could be brought because the alleged victims would not support police action.
However, Surrey Police did not tell any of them that other complaints had also been made.
The victims told a previous report into Savile’s abuse that if they had known, they probably would have been prepared to give evidence in court.
The force has admitted failings and this latest report – which took in evidence from 166 former Duncroft pupils – concludes its investigation.
In response to the report’s findings, Karen Froggatt, director at charity Victim Support said: “If those supposed to be protecting these vulnerable young people had spent less time judging their behaviour, and more time believing their cries for help, the dreadful abuse detailed in this report would have ended far sooner.”
Savile died in October 2011 without answering for five decades of prolific offending.
The former Top of the Pops presenter is thought to have committed more than 200 offences across 28 police force areas, including 34 rapes.
Three-quarters of his victims were under 18, with 28 alleged victims under 10-years-old.
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