A massive global search operation was launched and it took 10 days to locate the boys, who had taken shelter on a dry slope deep in the complex.
Narongsak also said Monday that the condition of the five remaining people in the cave is "still good".
The second operation started at 11 a.m. local time Monday.
Image: A crowd watches as ambulances transport some of the rescued boysA crowd watches as ambulances transport some of the rescued boys to a hospital on Monday.Lauren DeCicca / Getty Images Three more boys were carried out in the hours that followed, according to a post on the Thai navy SEALs Facebook page.
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.
British divers found the 13, huddled on a muddy bank in a partly flooded chamber several kilometres inside the complex, on Monday last week.
Bursts of heavy rain soaked the Tham Luang Cave area in Thailand's northern Chiang Rai province overnight, increasing the risks in what has been called a "war with water and time" to save the boys.
The rescued boys are likely to stay in Chiang Rai hospital for 48 hours undergoing medical tests to check their health after they swam through dirty floodwaters to exit the cave.
The soccer team had been trapped in the underground cave network since June 23, with flash floods causing water levels to rise, making it hard to get out.
After rescuing the four boys, the officials paused the operation for at least 10 hours to fill the oxygen tanks that were used.
Four boys were previously pulled from the cave on Sunday.
The four boys rescued Sunday were receiving medical treatment at a hospital in Chiang Rai, around 35 miles away.
In comments released by the government, Interior Minister Anupong Paojinda said officials were meeting Monday morning about the next stage of the operation and how to extract the remaining nine people from the cave in the country's north.
But after the first four emerged late on Sunday afternoon, hopes began to rise of a fairy tale ending to the ordeal.
All eight boys were rescued by divers using guide ropes and shared oxygen tanks.
All those rescued from the Luang Khun Nam Nang Non cave are to be flown or transported by ambulance to the Chiang Rai Prachanukroh Hospital. The boys are being quarantined due to the fear of infection.
"In the next few hours we will have good news", he said.
"I'm hoping for good news". Soldiers, medics, engineers and volunteers in yellow shirts milled around but the mood was relaxed. Seven boys remain trapped inside with their coach. On top of all the concerns inside the cave, Thailand's rainy season is looming.
Osotthanakorn said the boys were hungry and asking for a Thai dish of beef with basil and chilli.
It appears that the team were going into the cave as part of an initiation ceremony which tasked the boys with reaching the end of the cave and returning having written their names on the far wall.
The president of soccer's governing body, FIFA, has invited the boys to the World Cup Final in Moscow on Sunday if they make it out in time.