It comes after police opened an investigation into four other deaths at the hospital, including that of ten-year-old Milly Main.
A case note review published earlier this year found the infection listed on Milly Main's death certificate was “probably” related to the hospital environment.
Infection from contaminated water at the hospital was an “important contributory factor" to the death of another child, the report found.
The second child’s family has not been contacted or made aware of the circumstances surrounding their child’s death, Mr Sarwar said.
He has now reported the case to the Procurator Fiscal, he said.
Mr Sarwar told the Daily Record: “At the request of the Crown, I have now reported this second tragic case to the Procurator Fiscal.
“This should be a criminal investigation, just as rightly there is an investigation into Milly Main’s death.
“We know from an independent inquiry that the second child’s death was linked to the failings at the hospital, so it’s extraordinary that this was not already referred to the police.
“If they had died anywhere else, it would have been treated as a crime scene.“The health board management has known the circumstances of the child’s death since 2017 but did nothing and failed to contact them.”
An inquiry into the constructions of the QEUH and Royal Hospital for Children and Young People in Edinburgh continues.
The Scottish Hospitals Inquiry heard evidence from Milly Main’s mother, Kimberly Darroch, last week.
Ms Darroch called for the QEUH to be closed, and labelled her child’s death “murder”.
Milly, aged 10, died in 2017 following an infection linked to water. She was in remission from cancer following treatment at the hospital.
“My view is that the hospital should be closed. I don't think it's safe," Ms Darroch told the inquiry.
"I feel like the health board need to be punished for all of this. In my eyes, what happened to my daughter is murder.”
A Scottish Government spokesperson said: “Decisions on what the Crown Office investigates are wholly a matter for the Crown Office, which is independent of Ministers.”
The inquiry will take evidence from health boards at a later stage.