Five of those held were Napoli supporters and the other three were visiting fans from Germany, Alessandro Giuliano said during a news conference.
Napoli won the match 3-0, passing into the Champions League quarter-finals for the first time.
The Italian authorities banned fans living in Frankfurt from attending the stadium for the match amid concerns about possible violence, but a group of the German club's supporters made the trip to Italy anyway.
Footage showed supporters clashing with anti-riot forces and a police car in flames, with objects being tossed towards the police while overturned tables and chairs lay smashed on the ground.
Italian media said home supporters threw stones and bottles at buses carrying the Eintracht fans, damaging one window. Some 600 of them travelled from Germany, where Napoli won the first leg 2-0.
"These are not fans, they are criminals, thugs. I wonder if they would make the same mess in Germany?" Italy's Deputy Prime Minister Matteo Salvini wrote on Instagram, while expressing his "full support" for police on the ground.
Fans also clashed prior to the first leg in Frankfurt.
Eintracht were informed last week of Italy's decision not to allow their fans into the stadium, with the club reacting angrily and calling the decision "an unacceptable interference".
