Narongsak said earlier on Sunday the mission may take three or four days to complete. The boys were transferred by helicopter and ambulance to a hospital where they are being treated, but officials did not disclose details of their condition.
Interior Minister Anupong Paojinda said the same divers who took part in Sunday's rescue will return to extricate the others as they know the cave conditions and what to do.
As of this writing, four of the boys have already been rescued but operations have been put on hold to restock on supplies.
Fears of declining oxygen in the cave where the boys and their coach had holed up were eased when rescuers fed an air pipe into the area, accompanied by medics and expert divers.
Eight boys and the coach remain trapped, and the ordeal has been a closely watched news story around the globe.
Since the footage was taken expert divers have rescued four of the 12 boys. "The boys are ready to face any challenges", rescue chief Narongsak Osottanakorn told reporters near the cave site on Sunday morning.
Officials have said storms forecast for Chiang Rai province in Thailand's far north had factored into their decision to go ahead with a complicated and unsafe plan to have the boys and their coach dive out of the cave.
She said members of the 353rd U.S. Special Operations Group and U.S. military pararescue teams, survival specialists, divers and medical personnel are on the ground in support of the Thai-led global rescue effort.
Heavy monsoon rains lashed the mountainous Chiang Rai region, where the Tham Luang Nang Non cave is situated, for several hours overnight on Sunday.More news: Tropical Storm Chris to bring showers
It has assembled a team of internationally renowned divers and so far their expertise has brought four boys - thought to be the weakest - back to their families.
But for many in Thailand, Ekapol, who left his life in the monkhood three years ago and joined the Wild Boars as an assistant coach soon after, is an nearly divine force, sent to protect the boys as they go through this ordeal.
In an indication of how risky the journey can be, a former Thai navy diver died in the caves on Friday.
Meanwhile, a letter of apology from the coach of the football team was delivered to the boys' parents via the cave divers.
Thailand's Meteorological Department said there was a 60 per cent chance of rain with more thunderstorms forecast throughout the week. Reuters news agency reported that six boys have been rescued, citing a senior member of the medical team involved in the rescue. "If the oxygen levels are down and the carbon dioxide levels are up, then you can get too much carbon dioxide in your blood".
On Sunday, Musk tweeted a series of videos showing a team of divers testing the pod in the swimming pool at Palisades Charter High School.
The boys, aged between 11 and 16, went missing with their 25-year-old coach after soccer practice on June 23, setting out on an adventure to explore the cave complex, reportedly for a picnic to celebrate one of their birthdays, before being trapped by rising waters from bursts of monsoon rain.
Handwritten notes by the children trapped in a cave read, from the top: 'Please don't worry Dad and Mom, Biw has been gone for two weeks, I will help mom sell her goods every free day, I'll quickly make my way.' The message below reads: 'Please don't worry, be happy.
"I confirmed the factors [affecting the mission] are as good as yesterday. I want to say thanks for all the support and I want to apologize to the parents", he wrote.