Agnelli was also fined 20,000 euros ($23,770) while champions Juventus copped a fine of 300,000 euros ($356,000).
Agnelli was accused of helping sell tickets to ultras fans, several of whom have links with organized crime. The tickets were then resold for a huge profit.
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Three other club officials were also banned for a year and fined 20,000 euros by the FIGC who "partially accepted" the demands of prosecutor Giuseppe Pecoraro.
Pecoraro had requested a 30-month suspension and 50,000 euro ($59,000) fine against Agnelli. He had also demanded two Juventus matches be played behind closed doors and a European extension of the ban.
Agnelli - who was recently appointed president of the European Club Association (ECA) - has denied dealing with Rocco Dominello, a supporter very close to the Calabria-based N'drangheta mafia who was sentenced to more than seven years in prison in this case, but had recalled meeting him.
Juventus is accused of having ceded blocks of tickets to ultras group when it is not allowed to sell more than four at a time. The six-time defending Serie A champions had said that they would not request Agnelli to step down if he were found guilty.