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.
The pro-Roma group Associazione 21 Luglio, which regularly condemns discrimination and violence against Roma and Sinti in Italy - including repeated cases of local residents attacking camps with Molotov cocktails - expressed "pain" at the deaths.
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. According to a 2015 Pew Research Centre survey, anti-Roma sentiment is significantly stronger in Italy than almost any other European country.
The intolerance is fuelled by inflammatory comments by politicians on both the left and right quick to paint Roma as crooks.
The leader of the far-right Northern League, Matteo Salvini, is one of the most vocal, having previously said Roma camps should be bulldozed, and earlier this year offering legal support to two supermarket employees after they locked two Roma women in a recycling container and shared video of the scene online.
And though the Catholic Church has been praised for attempts to reach out to the Roma and Sinti communities, some campaigners have accused the pope of reinforcing stereotypes.