The striker was part of a co-ownership deal, but the Giallorossi won the blind auction with an offer of what is believed to be between 4.9m and 5.3m.
"We are very happy, because finally Fabio has made a dream come true, which is to become a Roma player completely," said agent Marco De Marchi on TMW.
"We hoped the two clubs would reach an agreement so we wouldn't have the adrenaline rush of the auction, but that's the way it went.
"We're aware that Roma sent an important signal in making the move permanent and consequently the lad is truly happy with this ending.
"He will do everything to repay the Giallorossi for their faith in him and become an important element for Roma."