He scored 24 goals and contributed eight assists in 36 matches last season for the Catalonian club who finished third after being title contenders for most of the campaign.
The details of the transfer were not disclosed, but Italian media reported the capital club will pay about 38 million euros ($41.48 million) for Dovbyk.
Manager Daniele De Rossi had been looking for more attacking options for next season after Belgian Romelu Lukaku returned to Chelsea following a loan spell.
Roma signed Argentine winger Matias Soule on Tuesday.