Arthur Cave was found with life-threatening injuries at the bottom of the Ovingdean Gap cliff at 6 p.m., and died of his injuries later that night in a nearby hospital, according to Sussex Police
His death is not being treated as suspicious, police said.
Nick Cave and his wife, British model Susie Bick, released a statement confirming the death of their son, who they called their "beautiful, happy, loving boy.
"We ask that we be given the privacy our family needs to grieve at this difficult time."
Arthur Cave is survived by a twin brother, Earl. The family lives in Brighton.
An Australian-born musician and occasional actor, Nick Cave is probably best known as the deep-voiced frontman for Nick Cave and the Bad Seeds, whose gothic songs have a brooding intensity.
Cave, 57, has also worked as a composer for such films as "The Road" and "The Assassination of Jesse James by the Coward Robert Ford" and has acted in a handful of movies, including "Wings of Desire" and "Johnny Suede."
Arthur and Earl Cave appeared briefly in "20,000 Days on Earth," a 2014 documentary about their father.
Nick, Susie and Earl Cave on Wednesday visited the edge of the cliff where mourners have left flowers and written tributes, according to The Daily Mail
Scores of fans also left condolences on Nick Cave's Facebook page
"Nick, your music has been an important part of my life for years and helped me get through some dark times. I am so sorry for your family's terrible loss," wrote one man. "Please know that all around the world people are thinking of you with love and sorrow in their hearts."