Had the call been given a grade one status, officers would have been sent to the address on a target time of ten minutes. The inquest heard that on the day the children were killed staff had been "overwhelmed" by calls.
An Independent Complaints Commission (IPCC) investigation had raised concerns about what was happening on the police control room that day.
A report in 2010 found "basic errors" in call handling and a "lack of resources" at the disposal of radio operators.
Coroner Nigel Meadows recorded a narrative verdict saying the children were unlawfully killed but that police neglect had contributed to their deaths.
He said: "I think it's entirely reasonable to conclude that probably, in the light of evidence, the lives of the children could had been saved."
Afterwards, the family's solicitor Alex Preston said: "The police had the opportunity to intervene and to take the boys and the mother to safety.
"The family are very pleased with the coroner's verdict that has held the police to account."
Dawn Copley, assistant chief constable of Greater Manchester Police (GMP), said she was "puzzled" by the verdict as two IPCC investigations had found no evidence of any misconduct in the way in which calls were handled.
She added: "GMP is asked to deal with an increasing number of calls to and from people who are suffering from mental illness.
"We are urgently talking to the health service and the the Home Office about our additional responsibilities in these cases and how best to deal with and protect people who are vulnerable."