An update on their Facebook page read: "12 boars and their coach leave the cave and everyone is safe".
Information on the rescue operation, the health of those rescued - and their identities - has been tightly guarded by Thai authorities.
They were found alive on July 2 after 10 days of being trapped in the flooded cave complex.
The optimistic mood was in contrast to the saddened atmosphere at the cave last week, after former Thai Navy SEAL Saman Kunan, 37, died while laying air tanks underground on Thursday.
Last week, Musk tweeted that he was sending a team of SpaceX and Boring Company engineers to Chiang Rai in northern Thailand on Saturday to help rescue efforts, supplying technology such as Tesla Powerwall batteries and pumps for removing water.
Rescued schoolboys are moved from a military helicopter to an awaiting ambulance at a military airport in Chiang Rai, Thailand, July 9, 2018.
The footballers still awaiting rescue are about two kilometers further in, at a point very hard to access.
Asked whether the remaining four "Wild Boars" team members and their coach would come out at the same time in the next rescue effort, he said: "It depends on the plan".
Premier Prayut visited the rescue base on Monday night to deliver his congratulations to all those involved, but also to offer a note of caution. Tuesday's operation began just after 10am.
The parents are able to see their kids through a glass window, but they are not allowed to make physical contact with them because docs are concern about infection.
And, Foisey said, the boys do have one thing going for them: "I'm sure these kids were very determined to get home".
Rescue leader Narongsak Osatanakorn said at a sudden press conference on Tuesday that the mission began at 10.08 a.m. local time and that everyone in the cave is expected to be rescued today.
He said a medic and three Seals in the cave, who have been looking after those trapped, will also come out.
"We have helped four more children today", said rescue chief Narongsak Osottanakorn.
Officials said all eight of the boys rescued so far were "good physically and mentally", though two of the four brought out on Monday had swollen lungs.
Still, health officials were wary of possible infections "because we have never experienced this kind of issue from a deep cave".
He said, "the kids are footballers so they have high immune systems".
Dr Jesada said the first group taken out, on Sunday, were aged 14 to 16 and the second group, taken out on Monday, were aged 12 to 14.
Four boys and their soccer coach remain in the cave.
After divers brought out four of the boys Monday evening, authorities indicated the rescue operation would continue for a third day.
Weather forecasters warned heavy rain could hit the area through the week. Monday's mission took about nine hours, two fewer than Sunday's.