The victims, aged four, eight and 20, were sleeping with their parents and eight other siblings when the van went up in flames.
Images from a surveillance camera show a man throwing a bottle against the vehicle before the blaze, investigators said.
It was not immediately clear if it was a xenophobic act or someone from another traveller family settling a score. Surviving family members had reported being threatened by locals in recent days, according to media reports.
President Sergio Mattarella denounced the "horrible crime", echoing Rome mayor Virginia Raggi's sorrow, and Pope Francis sent his chaplain to comfort the bereaved family and bring them "concrete assistance", the Vatican said.
Although over half the 170,000 or so Roma and Sinti people in Italy are Italian citizens with regular jobs and houses, hate crimes against the poorest strata are rife.
The intolerance is fuelled by inflammatory comments by politicians on both the left and right quick to paint Roma as crooks.
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