Australian-born Hania Ahmed and her family were visiting a relative at Chakwal, in Pakistan's Punjab province, when they were robbed while in their rental car late on Wednesday night (local time).
In a lengthy statement posted to X on Monday morning (Australian time), Punjab Police confirmed Crime Control Department officers intervened after the Perth family was taken hostage at gunpoint.
The suspects fired at a police officer, leading to an exchange of gunfire.
"In the ensuing chaos, the officer involved mistakenly assessed that the suspects were attempting to flee in the victims' vehicle and discharged his weapon," the statement said.
"This erroneous decision resulted in the tragic death of ... Hania and injuries to her father and brother."
The girl's mother was unharmed in the shooting.
The robbers fled on a motorcycle and were killed in a separate encounter with police on Thursday night (local time), Pakistani English-language news outlet Dawn reported.
The officer who opened fired on the Australian family's car completely deviated with established standard operating procedures and legal principles governing the use of force, police said.
He has been suspended from duty, formally arrested and faced court, where he was remanded to judicial custody in jail.
His weapon and spent shell casings are among forensic evidence that has been collected and sent for processing.
Pakistani police have promised a "thorough, impartial investigation to ensure that justice is served".
"We recognise that our primary duty is the protection of human life, and any failure to adhere to the principle of 'minimum force' is treated with the highest level of legal and departmental accountability," local police said.
"We are deeply saddened by this tragedy.
"While our personnel operate in high-risk environments, there is no justification for a departure from our protocols."
Hania was in year four at Australian Islamic College at Kewdale in Perth's east.
The school, where her older brother attends grade five, held a special assembly on Friday afternoon with students and has offered counselling services.
Executive principal Abdullah Khan remembered Hania as a "happy, bubbly, friendly girl".
"She had lots of friends, was very social and was liked by peers and teachers," he told AAP.
Mr Khan said he spoke with Hania's father, Adeel, on Sunday and communicated the school and community would wrap its arms around the family upon their return.
"He is devastated," he said.
"He's recovered well but the brother is still hospitalised and going through the treatment."
It might take more than month for the boy to be well enough to travel back to Australia, Mr Khan said.
Australia's Department of Foreign Affairs and Trade is providing consular assistance to the family.
WA Attorney-General Tony Buti on Sunday described the incident as a tragic case of "mistaken identity".
"Our thoughts and concerns are with the family and the Pakistani community that live in Western Australia," he said.
The Pakistani Association of Western Australia said the Ahmed family had, for now, requested privacy.
"We are in contact with family as they are passing through this hard time," the community group said.
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