Twenty-eight arrests were made while 13 police officers were injured in the clashes, which started on Wednesday and continued right up until kickoff on Thursday. Feyenoord fans also caused damage to historic monuments in Piazza di Spagna while disturbing the peace throughout the Eternal City.
"What happened is a sign of barbarism and uncivil behaviour," Renzi told Rai Due radio in quotes reported by the ANSA news agency. "We do not go abroad to embarrass ourselves like this. We are going to punish those involved severely. It's an absolute insult to civilisation."
The fixture was regarded as "high risk" after the draw was made, but the authorities struggled to cope.
Renzi added: "If we underestimated things ourselves then we will look into that, but our priority is to send them home and condemn those responsible.
"It is not acceptable or imaginable that first last night and then today they invade symbolic parts of the city and hold it to ransom.
"It's unacceptable. I expect apologies from Feyenoord."
A spokesperson for the Dutch ambassador, Aart Heering, issued an apology from the Dutch embassy, saying: "Football should be a festival and not a place for violence. The Italian authorities can expect total collaboration and dedication from Holland to ensure that those guilty are punished. We are with the police on this."
There were no reports of trouble after the game, although the Feyenoord fans were kept inside the Stadio Olimpico for an hour after the final whistle before being escorted to the railway station and airport.