The boys separately emerged from Tham Luang Nang Non cave in Chiang Rai Province between 5:30 pm and 6 pm local time.
On Sunday four members of the "Wild Boar" team were successfully brought out from the cave, after authorities decided they had to rush ahead with a rescue operation to beat monsoon rains.
It's unclear how long the complex and risky rescue effort will take, but it could be days.
Sam Teller, the spokesperson for the Boring Company, said four company engineers were "offering support in any way the government deems useful". The rescued boys are still being held in quarantine for testing and have not been able to have direct contact with their parents yet.
Authorities temporarily stopped rescue efforts on Monday to replenish air tanks along the cave's perilous exit course. All four were rushed to a hospital 65km away in Chiang Rai.
Four boys and their coach remain in the cave.
Musk tweeted early Saturday that he was working with a team from his Space X rocket company to build a "tiny kid-size submarine" to transport the children.
"It is an extremely risky undertaking", Bill Whitehouse, vice-chairman of the British Cave Rescue Council, told AFP, adding "poor visibility in the water, tortuous passages to get through, constrictions" were some of the major obstacles divers face. Rescuers have been unable to extend a hose pumping oxygen all the way to where the boys are, but have brought them some oxygen tanks. Their condition was not immediately clear.
Thirteen foreign divers and five Thai divers were leading the rescue, with each boy accompanied by two divers, officials said.
The operation began at 10 a.m.
Their escape led to an explosion of jubilation on social media in Thailand as the rescued boys were rushed to hospital.
The BBC's Helier Cheung has footage of an ambulance seen leaving the cave.
Between four to six boys have been rescued in the first phase of the operation to free the trapped soccer team, Thai officials told reporters.
Nine people remained trapped in the cave, including the team's coach, as of Monday morning. They had been missing for 10 days before they were discovered.
Officials said details of the rescue operation would be revealed on Wednesday.
"He added: "They can not decide how many of them will be able to come out for the first operation".
One cave explorer who has been inside the cave complex described it as a "labyrinth", adding it was much more hard to navigate than any other he had experienced.
Looming rain was one of the main enemies of the operation, threatening to flood the cave complex in mountainous northern Thailand, although a bewildering array of other dangers could also doom their safe return.
Authorities have said it takes roughly 11 hours to do a round-trip from the cave entrance to where the boys are huddled on a muddy bank.
"We need to refill", Mr Narongsak said. That's the name of the young boys' team.
The most risky part of the journey out of the labyrinth cave system is the first kilometer, during which they are required to squeeze through a narrow flooded channel.
Narongsak said some are well enough to ask for solid food. Having completed this section, the boys are then handed over to separate, specialist rescue teams, who help assist them through the remainder of the cave, much of which they can wade through.
A helicopter flew some of the boys to the nearby city of Chiang Rai where they were taken by ambulance to hospital. This was earlier than what was expected if conditions were ideal. The rescue mission was made harder by the high diving skills required to navigate the caves.
"If we wait and the rain comes in the next few days we will be exhausted again from pumping and our readiness would drop". "We have tried all possible ways to get the boys out, but sometimes we can't win over nature".
A Thai governor says the operation to bring out 12 schoolboys and their soccer coach from deep inside a cave where they have been trapped for two weeks has begun.
Narongsak Osottanakorn, who is in charge of the rescue mission, said Saturday conditions were now "perfect", BBC News reports. The second boy came out 10 minutes later. For the last several days, crews have been trying to drain enough water out of the cave to make the rescue easier.
Mild weather and a break in the rain had lowered the water to the lowest levels since the boys were found.