The rescue got off to a good start, with the first boy coming out at 5.40pm local time (11.40am United Kingdom time).
Rescue divers found the 12 teenage soccer players and their coach alive on July 3 after they went missing in a Thai cave 10 days earlier.
The rescue operation has been put on hold as all the air tanks installed along the evacuation route have been used up and need replacing, according to Narongsak.
Governor of Chiang Rai province, Narongsak Osatanakorn, said: "Today is D-Day".
When the boys are freed, they will immediately be taken to Chiangrai Prachanukroh hospital, around 60 miles from the cave site.
"Our operation today is more successful than what we've hoped for", he added.
The governor has called the rescue a "war with water and time". "We have tried all possible ways to get the boys out, but sometimes we can't win over nature". Osottanakorn said they will resume in 10 to 20 hours because "we've used all the oxygen".
Maj. Gen. Chalongchai Chaiyakam said the 13 "will continuously come out in approximately two to four days, which all may change depending on weather and water conditions", The Associated Press earlier reported. The boys urged relatives "not to worry" and asked for their favourite food once they were safely evacuated, in notes handed to divers.
One Thai navy SEAL died just over two days ago while involved in rescue efforts. According to the AP, the caption on the photo refers to the name of the boys' soccer team and says "We, the Thai team and the global team, will bring the Wild Boars home".
Reporters are ordered to move 2 kilometres back as final rescue preparations began on Sunday at Tham Luang cave in Mae Sai district, Chiang Rai.
Depleting oxygen levels in the cave have also been a cause for concern over the past week.
The boys, ages 11 to 16, and their 25-year-old coach became stranded when they went exploring in the cave after a practice game.
Monsoon flooding had initially cut off their escape, and rescuers were unable to find them for more than a week. "I promise I will take care of the kids", he said.
It is not yet known whether the boys swum out, or if they were sedated and delivered by the rescuers.
"Night loves Dad and Mom and brother, don't worry about me", wrote another boy in the Thai manner of referring to one's self in the third person.