The shootings took place on the streets before Saturday night's match between Napoli and Fiorentina and set up ugly scenes inside Rome's Olympic Stadium, where supporters threw flares and smoke bombs and delayed the start of the final.
Police said on Sunday they had arrested Daniele De Santis, a 48 year-old Roma supporter they said had insulted a group of Napoli fans and, when they responded, shot and wounded three of them with a pistol.
De Santis is charged with attempted murder while the three Napoli supporters are also under arrest. They include Ciro Esposito who is in a critical condition in hospital after a bullet hit him in the chest.
Commentators said the pre-match violence and the behaviour in the stadium, where supporters' boos and whistles drowned out the national anthem, underlined the decline of Italian football and the failure of its administrators.
Officials and politicians condemned the role given to Napoli's hard-core "Ultra" supporters, whose leader was consulted by the team's captain in the stadium before apparently giving his authorisation for the game to go ahead.
"What happened yesterday was the surrender of the country's institutions to the fans," said former national team coach Arrigo Sacchi.
"No-one wants to take any action because they are afraid of losing support, so the solutions are always put off and then forgotten about."
Repeated promises by politicians to crack down on football violence have proven ineffective, often resisted by the clubs themselves as well as their organised supporters.
Interior Minister Angelino Alfano said on Sunday he was considering introducing lifetime bans from stadiums for offenders, but many experts say Italy needs tougher criminal sanctions and an overhaul of its whole football culture to make it more family oriented.
Former Prime Minister Mario Monti suggested in 2012 that Italian professional football should be suspended for two or three years in the wake of a betting scandal, a call which was repeated by some commentators on Sunday.
In the 1990s Italy's showcase Serie A was widely seen as the most attractive league in Europe but now suffers from chronic crowd violence, dilapidated, half-empty stadiums, racism and deteriorating performances by its teams in European competitions.
Beppe Grillo, leader of the anti-establishment 5-Star Movement which is Italy's second largest party, wrote on his blog under the headline "The Republic is dead" that the cup final was like a funeral which confirmed the failure of the country's leaders and institutions.
The scenes in the stadium, watched by some 9 million television viewers, suggested little progress has been made in curbing the influence of the "Ultras" over the clubs.
The man accused of the shootings was among a group of Ultras who in 2004 prevented the derby between Roma and Lazio from being played after rumours - which turned out to be false - that a child had been run over by police outside the ground.
The sight of Napoli Ultras chief Gennaro De Tommaso apparently negotiating with team captain Marek Hamsik were reminiscent of the 2004 match when Roma's Ultras were consulted by captain Francesco Totti.
De Tommaso, whose photo was on the front pages of most newspapers on Sunday, wore a T-shirt calling for the release of a fellow Ultra in jail for the killing of a policeman at a Serie A match in Catania, Sicily in 2007.
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