Both players sustained the injuries in Saturday's 1-1 draw away to Atletico Madrid, before Barca were knocked off the top of La Liga by Sevilla on Sunday.
Rafinha, who has already had his career interrupted by two lengthy spells on the sidelines with serious knee injuries, will undergo surgery on his left knee next week, according to a club medical report.
Barca did not state how long the Brazil midfielder would be out for, although reports in the Spanish media said he was likely to miss the remainder of the campaign.
The club added that versatile defender Roberto, who also frequently plays in midfield, would be out for between three weeks and a month with an injury to his left hamstring.
It means he will miss up to four league games and the club's final Champions League group game at home to Tottenham Hotspur.