Northland Harbourmaster has been advised on the incident and an investigation is underway as to the cause of the incident, the Navy said.
The ship was on its way back from Waitangi following commemorations there during the long weekend.
The Rescue Co-ordination Centre got a distress signal about 7am followed by a mayday call shortly afterwards.
The mayday call was cancelled once the crew members had made it to safety.
Navy spokeswoman Lieutenant Commander Angela Barker said the 14m launch was now in Tutukaka marina.
She said an investigation into the incident was set to be launched once the vessel was back at Devonport Naval Base.
Kate Malcolm from Dive Tutukaka said the vessel, which she believed to be one used to transport dignitaries, has been tied to the beach off Marina Road.
"It has definitely taken on water. It's good they've got it into the harbour or it likely would have sunk."
An offshore patrol vessel has now arrived at the scene.
Ms Malcolm understood it was going to be towed out to sea by an offshore patrol vessel where the water would be pumped out.
"It's not a major issue ... maybe a little bit of an embarrassment for the Navy.
"We've already sunk two wrecks at Tutukaka. It would be nice to have three but we'll let them take this one back."
