Mr McGowan confirmed on Friday night his child was receiving ongoing treatment.
“The child is fully vaccinated, however, was in a serious condition as a result of the infection,” a statement from his office said.
“The child remains in hospital and is receiving ongoing treatment.
“The McGowan family asks for privacy regarding this situation at this difficult time.”
Mr McGowan has three children – two teenage sons and a young daughter.
The Premier revealed on Wednesday he was isolating at home for seven days after being deemed a close contact.
He tested pxjmtzywositive for Covid-19 on Friday.