Roma won the game 5-1 to top Group E of the Champions League, but the celebrations - on what was the club's return to the competition after a three-year absence - were marred by ugly scenes of violence before and during the match.
According to a web report on Italian sports daily Gazzetta dello Sport, one unnamed Russian was stabbed in the stomach and the arm and was transported to the city's Gemelli hospital.
Initial reports said the man's life was not in danger.
Another fan of the Russian Premier League side was taken to hospital with head wounds after being hit with what was thought to be a bottle.
Violence also erupted in the later stages of the game as Roma romped to a 5-0 lead.
Television pictures showed fans of both sides taunting and throwing missiles at each other, prompting the intervention of stewards.
When the Roma fans were cordoned behind a line of stewards, fans of CSKA began attacking stadium stewards, many of whom had to defend themselves from punches being thrown and what appeared to be iron bars before they moved back as police with guns and batons arrived.
The match was only briefly halted and shortly after resuming CSKA claimed their only goal of the game through Ahmed Musa.
Roma coach Rudi Garcia told Canal5 after the game: "It's not right. This is a football game and shouldn't turn into a battle.
"We should take steps to get rid of people who are involved in this kind of behaviour. Outside the stadium, there are police to deal with this kind of problem."
The Italian capital has witnessed several violent incidents over the years.
Earlier this year Napoli supporter Ciro Esposito died in hospital after being shot by hardline ultra fans of Roma as fans of Napoli and Fiorentina travelled to Rome for the Italian Cup final.