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The Giallorossi raced out into a three-goal first half advantage in the Turkish capital - as Jordan Veretout, Justin Kluivert and Edin Dzeko all got on the scoresheet.
And that was more than enough for Paulo Fonseca's side to secure a much-needed three points, and leave them knowing a win in the final group game at home to Wolfsberger AC will guarantee their place in the last-32 of the competition.
In truth Roma showed themselves to be a cut above at the Fatih Terim Stadium, although they needed a first-half penalty award to break the deadlock.
Mehmet Topal appeared to block Lorenzo Pellegrini's shot with his hand, and Veretout made no mistake from the spot.
Pellegrini then turned assist-maker twice, beautiful lobbed balls allowing Kluivert and Dzeko to get in behind the defence and score.
The win could have been enough to send the Giallorossi into the knockout stages, but Borussia Monchengladbach's 1-0 win in Austria means there is still some work to be done in two weeks' time at the Olimpico.
ROMA: Pau Lopez; Kolarov, Smalling, Mancini, Santon; Veretout, Diawara; Zaniolo, Pellegrini, Kluivert; Dzeko.
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