It is believed to be the first time foreign fans have travelled to Italy even when their team were not playing in the country and launched a surprise attack on their rivals.
The ambush by a group of men wearing black happened on Saturday evening near the capital's Olympic stadium after Roma's Serie A game against Empoli.
The hardcore Roma supporter group, the Fedayn, said the assailants were Serbian. Photographs posted on Telegram showed that they had seized a bag containing giant banners that the Fedayn display inside stadiums on matchday.
Italian police are looking into the incident and are braced for further potential trouble, a spokesperson said.
Red Star Belgrade fans have an alliance with radical supporters, or ultras, from the southern Italian team Napoli, who have a long-standing rivalry with Roma.
Roma and Napoli ultras clashed at an Italian motorway service area in January, earning fans from the two clubs a two-month ban for away games. Local media speculated that Saturday's raid might have been in response to the motorway bust-up.
Losing the team banners is a deep humiliation for ultra groups and Roma fans warned of major trouble if their insignia show up in Naples' Maradona stadium.
"If our banners are strung upside down in Naples, civil war will break out," one anonymous fan wrote on Facebook.
Ultras from northern Italian club Inter Milan issued a statement condemning the Red Star fans. "This (action) can dangerously shift the balance of dynamics linked to rivalry... We hope it does not create a precedent," it said.
By contrast, supporters from Roma's city rivals Lazio issued a statement to deny media reports that they had offered their sympathies over the stolen banners.
"This historic moment of shame that they have suffered has nothing to do with us," they wrote.
