Lorenzo Grossi gave Alberto De Rossi's side a first-half lead in the play-off clash, but the visitors scored twice in as many minutes shortly before the hour-mark to ultimately turn things around.
Roma had been reduced to 10-men at that point, after captain Eros De Santis was shown a second yellow card - although goalkeeper Lorenzo Crisanto saved the subsequent penalty kick.
But Monaco were awarded another one soon after, converted by Adrien Bongiovanni, before Moussa Sylla tapped home moments later to completely change the course of the tie.
Roma pushed for an equaliser after Monaco were also reduced to 10-men, but an equaliser proved elusive.
A third red card of the contest, this time for Stefano Ciavattini, completed a disappointing afternoon for the Giallorossi.
"We were up against a really tough side and they certainly showed that," De Rossi said afterwards. "I’m disappointed with the fact that we had chances, in the first half especially, but we didn’t take them as we usually would. We had many chances in the first half to break and score more, but we didn’t manage to do so."
He added: "[Monaco] really went for it, pushing their full backs high up the pitch and leaving only their centre backs to mark up. There were opportunities for us to break but we kept messing up the final pass. In the second half we weren’t quite savvy enough."
See more images from Tre Fontane in the gallery below.
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