Antonio, who was 14 when he fell ill, has now been given the all-clear after a gruelling course of treatment.
The former deputy prime minister and his wife Miriam Gonzalez Durantez spoke to the cancer charity Bloodwise about their experience of the disease.
Their eldest son was diagnosed with stage two Hodgkin's lymphoma in his neck and chest after a biopsy on a painless lump in September 2016.
Clegg told the charity that his first reaction 'was an overwhelming, if irrational, wish to take the cancer away from him and take it on ourselves.'
Antonio was treated in a teenage cancer unit, where he had four monthly cycles of chemotherapy, took a course of heavy steroids and was given around 20 tablets per day to take.
The Cleggs are campaigning for more research into kinder treatments that avoid detrimental side effects such as hair loss, vomiting, tiredness and immuno-deficiencies.
The teenager has now recovered and has regular checks to ensure the disease has not returned. His parents have called for support for Bloodwise's research into lymphoma recovery rates.
Clegg, who lost his Sheffield Hallam seat in June's general election, also has two younger sons with his Spanish wife Durantez.
Today the couple told ITV's Lorraine Kelly about the family's ordeal.
"We dealt with it by carrying on and trying to keep things as close to the routine that we had beforehand and also being very open," said Durantaz.
"The day that he was told, and I think that probably us telling him is one of the toughest things that we have ever done, the following day he went to school, he stood up and he told everybody 'I have cancer'.
"That's the way he dealt with it but other children and other families deal with it in a different way, you have to find your way."
Mr Clegg added:
"His lymphoma was all over his chest and his neck and he gets tested every three months, I think for a couple of years, so there is always a slight spike of anxiety with us every three months but basically he is on the road to recovery.
"Interestingly the thing he was most concerned about was sort of falling behind his classmates. His anxiety was more about keeping up with his classmates, keeping up at school. So it was very impressive actually."
He added the couple's other children, Alberto and Miguel, had taken the news well.
"Once they heard from us that he will be OK, again they are just so, so practical - just 'OK then'," he said.
Dr Alasdair Rankin, director of research at Bloodwise, said: "The reality is that one in five children diagnosed with the most common type of leukaemia still do not survive, and that those who do often experience devastating side effects both during and after treatment.
"This is simply not good enough. We need to save every child's life, make the treatment process much kinder and give them the life they would have had without cancer."