Over 30,000 fans from both clubs were estimated to have travelled to the capital for the final at Rome's Olympic Stadium.
However several reports, including one from Gazzetta dello Sport, suggested the clashes were caused by fans of Roma fighting with those of Napoli.
Gazzetta's website report said four men had been injured, one of whom was a plain clothes policeman.
Earlier, domestic news agency ANSA reported that three supporters had suffered gunshot wounds in a violent prelude to the final.
"These incidents are simply unacceptable," said Maurizio Beretta, the president of Lega Serie A which also organises the Cup competition.
"We appeal to the fans who are here for the final to treat it as it should be -- an evening of sport and football and not one that should give rise to episodes of violence."
No official reason was given for the delay in proceedings.
The match, due to kick off at 2100 local time (1900 GMT), eventually got under way 45 minutes late.
Discussions between dozens of officials took place just before kick-off when Napoli fans in the Curva Nord (North End) began throwing flares towards the pitch.
It is believed hardline 'ultra' fans of Napoli had demanded their players not play the game after hearing that one of their fans had been shot.
In an apparent bid to restore order, Napoli forward Marek Hamsik was eventually ushered over to speak to supporters.
The Slovakian spoke to several representatives of Napoli's support before returning pitchside.
Meanwhile, Italian Senate president Pietro Grosso said: "A football match shouldn't be about battles between two sets of supporters.
"We're here to see a final, to enjoy the game in a festive and sporting atmosphere. I'm saddened to see that this kind of thing is still going on."