Mr McGowan also tested positive for the virus after the child's admission to hospital and he is isolating at home.
"After a very worrying four days, the premier's child is now recovering at home with family," a government spokesman said, as reported by the Perth Now online news site.
"The McGowan family would like to thank everyone for their supportive and kind comments," the spokesman said.
"The McGowan family thanks the committed hospital staff for all their work and care."
According to a statement from the premier's office on Friday evening, "the child is fully vaccinated but was in a serious condition as a result of the infection".
Mr McGowan had initially returned a negative test after becoming a close contact.
In a statement, the premier said he had taken another PCR test on Thursday and it had come back positive.
"In accordance with the protocols, this will extend my quarantine period until I am hopefully cleared to leave home next Thursday afternoon, at the earliest.
"I will continue to quarantine and work from home over this period," the premier said.
He is scheduled to attend federal Labor's election campaign launch in Perth on May 1.
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