The 24-year-old from Penrhyn Bay, Conwy, and his fellow reservist died on the hottest day of the year in Wales.
A third soldier is seriously ill after the incident thought to have involved six soldiers collapsing.
Mr Roberts lived in London and was a maths teacher in Lewisham.
His father Kelvin, 53, and mother Margaret, 54, were said to be “heartbroken” by his death.
“We are being given the support of the military and anything we want to see will come through them,” said his father.
Mr Roberts, who had lived in London after leaving university, had served with the Territorial Army (TA) for more than five years.
He had taken part in exercises several times including a training mission in Texas with the American National Guard.
Some of it involved hot weather training in the desert aimed at replicating conditions in the Middle East.
He was a member of the 3rd Battalion Royal Anglian Regiment and is believed to have served as a reservist in Iraq and Afghanistan.
L/Cpl Roberts was described as being “the salt of the earth” by friends.
“CJ loved being in the TA and really wanted to go as far as he could. He was extremely dedicated,” said one.
“He was the salt of the earth and would have a go at anything. It is devastating to think how he died like that.”
L/Cpl Roberts graduated from Leicester University in 2010 with a degree in banking and finance and was a keen sportsman with a love of rugby.
He had taken part in a four-week trial for the TA SAS reservists ending in a 40-mile hike over the Brecon Beacons.
It would have been a supervised exercise, but it is likely it would have involved them acting on their own moving from checkpoint to checkpoint.
BBC security correspondent Frank Gardner said he understood the deaths occurred after the training and preparation phase and right at the start of the selection phase.
He also understood that there were extra water supplies available in the area in addition to the water the men were carrying.
An MoD statement released on Monday said: “It is with great sadness that the Ministry of Defence can confirm the death of L/Cpl Craig Roberts and another Army reserve soldier during a training exercise in the Brecon Beacons at the weekend.
“Our thoughts are with their families and friends at this difficult time.”
The Brecon Beacons is home to the Infantry Battle School and makes up one of Britain's largest military training areas.
The deaths occurred on the hottest day of the year so far in Wales, with temperatures reaching 30C (86F) in Powys.
Phil Speck, who was walking in the Beacons on Saturday, said he saw some soldiers struggling in the heat.
“We saw many soldiers across our walk – probably a dozen or so,” he said.
“To start with they looked fine. Towards the end of the day they were feeling the heat. One soldier spent quite a period of time recovering.”
The soldiers appeared to be in full kit and he said some “were clearly tired and exhausted”.
“When you see a soldier you kind of assume that they are fit enough enough to deal with the situation,” he added.
Gethin Havard, who farms near Pen y Fan – the highest mountain in south Wales – said he arrived at a car park below the mountain at about 17:00 BST and could see a soldier being treated by paramedics.
“I could see there were some soldiers huddled about 20 yards away,” he added.
“Another farmer told me he'd seen them carrying a soldier down. There didn't seem to be any panic.
“The medics tended to him for a little while and they put him into the ambulance.”
Mr Havard said one of the paramedics had “rushed over to get water from the toilets, but what they did with it, I don't know”.
He added that one of his daughters had seen a soldier being brought from the mountain with a bloody nose.
It is understood live ammunition was not involved in Saturday's exercise.
The MoD said it was working with Dyfed-Powys Police to investigate the incident.
A spokesman added there are no plans to change “routine exercises”.
Conservative MP Colonel Bob Stewart – a former British Army officer and member of the Defence Select Committee – said the soldiers would have been largely unsupervised and the symptoms would have come on very quickly.
“They were probably going cross country, up and down the mountains on their own, which makes it even more difficult to identify heat exhaustion or heat stroke because, of course, they've got no friends around them to do something when they collapse,” he explained.
In January Army captain Rob Carnegie was found dead on a snow-covered Corn Du mountain, south west of Pen y Fan.
It was thought Capt Carnegie had been taking part in a march in freezing conditions as part of a selection process for the special forces regiment.