Mr Gravatt died in July 2009 during the swine flu epidemic.
The joint statement notes that coroner Brandt Shortland, relying on an independent medical report by Dr Roger Reynolds, found shortcomings in the care Mr Gravatt received at the hospital.
"... [these included] that there were delays in treating Zachary, in considering the diagnosis of meningococcal disease and in referring Zachary to the department of critical care medicine.
"Auckland DHB has accepted these shortcomings as identified by Dr Reynolds and found by the coroner. Auckland DHB sincerely and unreservedly apologises to Dr [Lance] Gravatt [Zachary's father] and the Gravatt family for the shortcomings.
"Auckland DHB accepts that a number of aspects of the shortcomings represented substandard care of Zachary at Auckland City Hospital.
"The Gravatt family and the Auckland DHB do hold different views on whether the shortcomings in Zachary's care caused his death. The Gravatt family sincerely believes that these shortcomings resulted in Zachary's death, in the sense that Zachary could have survived with better treatment. The Gravatt family has suffered considerable distress as a result."
As part of the coroner's inquest, the DHB and the Gravatts agreed on measures to minimise the risk of the identified shortcomings occurring again.
"These changes have now been made and the lessons of Zachary's case have been taken up by the health sector," the statement said. "This is partly due to the tireless efforts of Dr Gravatt."