The supporters rampaged through Rome's famous Piazza di Spagna, injuring several police officers after throwing flares and other objects.
The square and the iconic Spanish Steps were left covered with beer bottles and other litter, while the recently-restored Barcaccia fountain was damaged.
"I have complained and asked for explanations from those responsible for public safety in this city, as to how they could have allowed precious and recently-restored monuments like the Barcaccia to become the target of violent actions," Rome mayor Ignazio Marino said.
"I am seriously angered. What is happening is unacceptable. I have spoken several times with the Prefect and we have sent specialists to assess the damage to the Barcaccia."
The Barcaccia fountain — which is nearly 400 years old — was only reopened in September after a year-long restoration project costing more than 200,000 euros ($227,000).
Shops in the area were also forced to close in fear of the fans, who at one point unfurled a banner which read "Europe be aware."
UEFA, the governing body of European soccer, will not take any action as the incident is outside its remit because it occurred away from the stadium.
On Wednesday, 23 Feyenoord fans were arrested after causing trouble in Rome's Campo de Fiori.
Roma hosts the Dutch club in the first leg of their Europa League last-32 match later Thursday.
Feyenoord was thrown out of the 2006-07 UEFA Cup after crowd trouble during a group stage match at Nancy.
The club also had to close a section of its De Kuip stadium for a Europa League playoff match in August, and pay a fine of 45,000 euros ($60,000) after racist behaviour by fans during a Champions League match in July against Besiktas of Turkey.