The Greece defender struck from close range after only six minutes at the Stadio Olimpico on Thursday and his second goal for the Eternal City outfit proved to be decisive.
Roma suffered their first defeat under Rudi Garcia when they were beaten 3-0 at Juve on Sunday, but got back on track to set up another clash with the Serie A leaders.
Radja Nainggolan was handed a debut by Garcia following his loan move from Cagliari and the midfielder came close to a goal, only to be denied by Vincenzo Fiorillo.
The Belgian still enjoyed a winning start to his time at the club as Serie A rivals Sampdoria were unable to fashion an equaliser.
Inter missed out on a potential last-eight clash against arch-rivals Milan, suffering a 1-0 defeat at Udinese in the other tie to be played on Thursday.
Brazilian forward Maicosuel struck 32 minutes in at Stadio Friuli and Inter were unable to respond despite bringing on Rodrigo Palacio and Ricky Alvarez, meaning Udinese will now face either Milan or Spezia in the quarter-finals
There was a positive for Inter coach Walter Mazzarri, though, as striker Diego Milito made his first start in almost a year and lasted 90 minutes following his recovery from knee and thigh injuries. Ruben Botta also made his Inter debut as a substitute after overcoming a knee problem.
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