The 15-year-old had suffered life-threatening injuries and was taken to hospital but died later that evening.
Cave and his wife Susie said: “Our son Arthur died on Tuesday evening. He was our beautiful, happy loving boy. We ask that we be given the privacy our family needs to grieve at this difficult time.”
Police said the death is not being treated as suspicious.
Arthur, who lived in Brighton, died at the Royal Sussex County Hospital. A coroner is now investigating his death.
Arthur was found unconscious by members of the public on the beach underneath Ovingdean Gap, an 18m-high cliff.
It is understood some walkers performed first aid on him before an air ambulance took the teenager to hospital, but he died later from his injuries.
Arthur’s friend Ed Bucknall was among a group who visited the scene of the tragedy. He said: “He was my best friend. I saw him every day and I can’t believe what’s happened. I was with him in the morning, he was round mine.
“He was one of the nicest guys you could meet, he was so funny and just so much fun. He never said a bad word about anyone.
“He loved skating, we always used to skate together. And now we can’t. I’m devastated, I can’t even explain it. I’m just broken. I’ve been distraught ever since I heard. I was with his brother Earl when I heard and I went and saw their dad.
‘I’m not sure what Arthur was doing at the time, I have no idea how he would have fallen. He was just such a nice guy, you couldn’t have asked for a nicer mate,” he said.
Cave is an Australian musician and songwriter, best known as being lead singer of the band Nick Cave And The Bad Seeds, an alternative rock group formed in Melbourne in 1983.
Last year, he and his wife appeared alongside Arthur and his twin brother Earl in 20,000 Days On Earth, a film documenting a fictitious 24 hours in the singer’s life.
The singer has two other sons, Luke and Jethro.
The family live in central Brighton.