West Yorkshire Police said a post-mortem examination “was not able to determine how she died” and that officers were continuing to investigate.
Police said they were called to a flat in Sheepridge Road at around 11:36pm on Monday night following a report a girl was unresponsive.
She died in hospital in the early hours of Tuesday morning, West Yorkshire Police said.
A 16-year-old boy was arrested on suspicion of murder and rape and remains in custody.
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Detective Chief Inspector Stacey Atkinson said: “We are continuing to conduct a number of inquiries today following the death of a 13-year-old girl following an incident at a flat on Sheepridge Road.”
She added: “A post mortem was not able to determine how she died, and we continue to treat her death as unexplained at this time, pending more inquiries.
“Those inquiries are complex and are likely to be lengthy as we work to fully understand the circumstances of the girl’s death.
“Specially trained officers are supporting her family at this time.”
The force confirmed the 16-year-old boy was arrested on suspicion of another offence of rape relating to a 16-year-old girl from a separate incident, elsewhere in the Kirklees district.
