The Associated Press reports an ambulance was seen leaving the area on Monday.
Authorities have been tightlipped about the progress of Monday's operation.
"As all of us said, the main thing is we still need more than 20 hours to get ready", Narongsak said.
The premier arrived late Friday at the mouth of Tham Luang cave.
Divers held the boys close to bring them out, and each had to wear an oxygen mask to enable normal breathing, authorities said.
All those rescued from the Luang Khun Nam Nang Non cave are to be flown or transported by ambulance to the Chiang Rai Prachanukroh Hospital.
Less than an hour earlier, an ambulance with flashing lights had left the cave complex, hours after the start of the second phase of an operation to rescue the soccer team.
But rescuing the remaining five people, which includes the boys' 25-year-old soccer coach, could take more than one operation, Narongsak said.
The billionaire entrepreneur has been tweeting proposals and offering his services to the Thai government in its efforts to extract the 12 boys and their soccer coach from a partially flooded cave.
Rescue operations on Monday were expected to end by 9 p.m. local time.
Rescue operations were stopped Sunday evening post the rescue of four boys to replace air tanks and supplies, and it resumed Monday morning.
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Narongsak said 13 foreign divers and five Thai navy SEALs took part in the key leg of Sunday's rescue: taking the boys from where they have been sheltering and through dark, tight and twisting passageways filled with muddy water and strong currents. We've got good news for you.
A fifth boy was brought out earlier on Monday, a navy official said. Previously, round trips through the cave network would take about 11 hours.
In a letter of his own, the coach, Ekapol Chanthawong, apologized to the boys' parents for the ordeal.
Onlookers watch and cheer during the rescue operation.
Interior Minister Anupong Paochinda told reporters the rescued boys were in good health in hospital but did not give further details.
There was no immediate word on their conditions. Each round trip from the rescue camp to the rocky shelf where the group is trapped takes about 11 hours.
None of the trapped boys has ever been diving before and some do not even know how to swim.
Workers have been laboring around the clock to pump water out of the cave, and authorities said Monday that heavy downpours overnight did not raise water levels inside.
Thailand's Meteorological Department said there was a 60 percent chance of rain Monday with thunderstorms forecast throughout the week. Musk has also shared a video of the submarine being tested in a swimming pool, which as per another tweet by the founder of SpaceX, can also be used in future space missions.
Officials have yet to confirm the identities of the four boys freed. Two divers were assigned to each child to help them navigate the risky, narrow passageways.
He said that the decision to try to rescue the boys and their coach was made because conditions inside the cave were the best they could hope for and that water levels were low enough after days of good weather that it was possible to walk through long stretches of the passage.