Police said the area was off-limits at the Solfatara Crater in Pozzuoli, one of 40 volcanoes in the Campi Flegrei area about 7 miles west of Naples.
Italian reports identified them as Massimiliano Carrer, Tiziana Zaramella and their son, Lorenzo.
Firefighters said the family, which were from Turin, appeared to have died from asphyxia, possibly because of hot gases emanating from the ground.
A view on Solfatara di Pozzuoli where three people died in the crater at Pozzuoli, Naples, Italy, 12 September 2017.
Their 11-year-old son ran through a safety barrier and went onto an unstable portion of the crater that is made of crumbling quicksand. It was a tragedy; they pulled out two bodies, then pulled us away. "I express my closeness from the community of Pozzuoli to the victims' family", he said.
Tourists are allowed to walk around the ground level Solfatara crater, which is dotted with trees, while admiring fenced off bubbling mud pools and fumeroles that spurt steam and sulphurous gases.
"I've been here for a long time and such a mischance has never happened", Armando Guerriero, proprietor of a bar situated close to the passage to the spring of gushing lava, told the ANSA news office. "We tried to calm him".
Solfatara's crater fields are scorching hot only a few inches below the surface and last erupted in 1198, the Daily Mail said.