The Brazil international had only been on the pitch for a few minutes when he was injured by a robust tackle from behind by Roma midfielder Radja Nainggolan.
Barca said on their website on Thursday Rafinha, the younger brother of Bayern Munich midfielder Thiago Alcantara, would have the operation in the next few days.
They did not say how long he might be sidelined but Spanish media reported he would miss the rest of the season.
Thiago, who plays for Spain and like Rafinha is a graduate of Barca's academy, needed two operations on a similar injury he sustained in March last year and missed the 2014 World Cup finals in Brazil.
Rafinha is not an automatic starter for Barca coach Luis Enrique but has been on good form in recent weeks and will be a significant loss for the Spanish and European champions.
Their next outing is a La Liga game at home to Levante on Sunday and they host Bayer Leverkusen in the Champions League at the end of the month.
Rafinha and Thiago are the sons of former Brazil midfielder Mazinho, who played in the 1994 World Cup-winning side.
(Reporting by Iain Rogers, editing by Amlan Chakraborty)
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