Six people were injured, one of them seriously, in what police have said was a racially-motivated attack.
The suspect, 28-year-old Luca Traini, had an Italian flag draped over his shoulders when he was arrested.
Witnesses said he had also made a fascist salute.
PM Paolo Gentiloni said: "The judiciary will make its decisions but one thing is certain: heinous crimes and criminal behaviour will be prosecuted and punished."
He added: "I trust the sense of responsibility of all political forces: criminal behaviour cannot have any ideological motivation - delinquents are delinquents."
The shooting on Saturday happened just days after a Nigerian man was arrested in connection with the death of an 18-year-old Italian woman.
Pamela Mastropietro's body was found dismembered and packed into two suitcases near Macerata.
Times journalist Tom Kington had told Sky News her murder had "created a huge wave of intolerance towards migrants locally, and has also been picked up by national right-wing politicians".
Mr Gentiloni warned: "The state will be particularly severe towards anyone thinking of fuelling a spiral of violence.
"Let's stop this risk, let's stop it right away, let's stop it together.
"Hatred and violence will not be able to divide us.
"The Italian people will know how to get together around the institutions and common values of the Republic."
The shootings came as Italy heads towards its national election on 4 March, where immigration has become a key issue.
More than 600,000 migrants - mainly from Africa - have reached Italy by boat over the past four years.
Matteo Salvini, head of the anti-migrant Northern League, has capitalised on Ms Mastropietro's murder in his campaign, and has vowed to deport 150,000 migrants in his first year of office if elected.
It has emerged that Traini once stood as a candidate for the right-wing party at last year's local elections but he did not receive any votes.