The team became stuck in a Tham Luang Nang Non cave on June 23, but the location wasn't discovered until July 2. Narongsak said the four were then taken to a hospital.
The names of the rescued boys were not released. After the boys emerge from the cave, they will be taken in an ambulance to be evacuated from the cave site via helicopter, provided weather conditions are safe for flying.
Narongsak had dubbed Sunday to be "D-day" as the complicated effort was launched in the morning.
They were trapped in the flooded cave for nine days before being found.
But experts have warned that extraction efforts will bear significant risk, underscored by the death of a retired Thai Navy SEAL early Friday morning when he ran out of oxygen during a dive.
Finnish volunteer diver Mikko Paasi, a long-term resident of Thailand, said the death of the Thai Navy SEAL had changed the mood on the ground and made real for rescuers just how risky the mission had become. All four boys were said to be in "perfect" condition, and safely transported to a hospital.
Officials identified the rescued boys as Mongkol Boonpiam, 13, Nattawut Thakamsong and Prajak Sutham, both 14 and Pipat Bodhi, 15, according to reports. Authorities had said that incoming monsoon rains that could send water levels in the cave rising, coupled with falling oxygen levels in the enclosed space, added to the urgency of getting those trapped out.
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The rescue mission is proving to be hard for divers whose efforts are continually hampered by rising water that has filled sections of the cave, often forcing them to stop. Operation chief Narongsak Osottanakorn said a total of 90 divers would be involved in the rescue mission, with 40 divers from Thailand, and 50 from overseas, including the United States, Australia, China, and Europe, according to the Washington Post.
The evacuation came several hours after 13 worldwide cave diving experts and five Thai Navy SEALs entered the cave in an attempt to get the boys out one by one as they navigate through flooded and narrow tunnels. The team and coach were exploring the cave after a practice game when heavy rainfall and flooding cut off their escape route out of the cave and prevented rescuers from finding them for nearly 10 days.
The ordeal has riveted Thailand and captured the world's attention.
The second rescue is again expected to involve a large foreign contingent - with 50 foreign divers, including Australian and British experts, assisting Thai authorities.
Before the current rescue mission began, Elon Musk tweeted that he was designing an "escape pod" built from "the liquid oxygen transfer tube" of a SpaceX Falcon rocket that could be used as a submarine.
While the initial strategy was to extract the strongest boys first, that was revised and the weakest were taken out after Dr Harris assessed their strength and health, Thai media reported.
In one of the videos of testing that Musk put on Twitter, a man can be seen emerging from the metallic cylinder. "Very fearless and talented people!" he tweeted. One boy's note said: "I'm doing fine, but the air is a little cold, but don't worry". I've told Yod to get ready to take me out for fried chicken.
"Don't be anxious", wrote another boy, Mick.