The Giallorossi dominated proceedings at the Stadio Olimpico but were once again thwarted by an incredible goalkeeping display, this time from Torino's Vanja Milinkovic-Savic.
Lorenzo De Silvestri and Simone Edera scored against the run of play for the visitors, with Patrik Schick's first goal for the club coming late on as Eusebio Di Francesco's side poured forward in search of a way back into the game they had largely dominated.
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Time and time again, however, they were thwarted by Milinkovic-Savic - who saved a second-half Edin Dzeko penalty and made a number of eye-catching stops from the likes of Cengiz Under and Diego Perotti.
And, when the Serbian was beaten, it was the woodwork invariably came to Torino's aid - with Stephan El Shaarawy and Schick both denied by the frame of the goal during the 90 minutes.
Torino will now go on to face either Juventus or Genoa in the quarter-finals of the competition.
Kevin Strootman captained the side as goalkeeper Lukasz Skorupski made his first appearance of the campaign - while full-back Emerson Palmieri started for the first time since suffering a serious knee injury at the end of last season.
Schick made his third start in a row in attack, while the back four featured four different players from the unit that kept a clean sheet against Cagliari last Saturday.
16' - A sustained spell of Roma pressure ends with Gonalons firing an ambitious shot high and wide from the edge of the box.
23' - GREAT SAVE! Edera swings in an inviting free-kick that De Silvestri meets powerfully inside the box, but Skorupski throws himself to his right and claws the ball away!
31' - CROSSBAR! SO CLOSE! El Shaarawy hits an incredible curling drive from 30 yards that clatters off the underside of the bar, down onto the goalline, and then bounces clear!
33' - Emerson fires wide from just inside the box.
36' - CHANCE! Glorious - albeit difficult - opening for Cengiz, who smashes a first-time half-volley over the crossbar from 10 yards out.
39' - Goal. Torino lead. Belotti's free header from a corner is saved by Skorupski, but De Silvestri finds the back of the net from the rebound. 0-1.
41' - Nearly an immediate response from Roma. Gerson feeds Cengiz, who forces a good save from Milinkovic-Savic.
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48' - CHANCE! El Shaarawy meets Gonalons' cross at the back post ... but Milinkovic-Savic makes a vital block!
59' - Chance for Torino on the counter. Acquah feeds Edera but he fires over as Skorupski rushes from his line to try and make the stop.
62' - CHANCE! Schick meets the corner but directs his header straight at Milinkovic-Savic!
64' - Good spell from Roma. Perotti forces another good low stop from Milinkovic-Savic, this time after a nice crossfield ball from El Shaarawy.
65' - Another chance! This time it is Strootman who cannot steer his first-time shot on target after being left free inside the box...
66' - Milinkovic-Savic makes another good save, this time to parry away Gerson's low drive from distance.
67' - POST! Schick beats Milinkovic-Savic to Strootman's inswinging free-kick... but the ball hits the outside of the far post and goes wide!
72' - Goal. Torino score again. Edera ghosts in at the far post to turn home the inswinging cross. 0-2.
76' - Penalty for Roma. Schick is hauled back inside the box as he tries to shoot...
77' - Penalty saved. Dzeko hits a powerful strike but Milinkovic-Savic dives to his right to make the stop.
82' - Another save from Milinkovic-Savic to deny El Shaarawy.
85' - GOAL. Roma get one back. Patrik Schick takes a touch and lashes home brilliantly from inside the box - his first goal for Roma and a lifeline for Roma in this game! 1-2!
88' - Milinkovic-Savic dives to his right to keep out Dzeko's powerful low drive.
89' - Dzeko meets an inviting cross from Peres but cannot steer his header on target!
93'+4 - Schick turns and hits a shot but it's deflected wide. The corner comes to nothing...
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“It's a shame because we created loads of chances tonight. Perhaps we were found a bit wanting on set pieces. We got distracted and lost our men – that's why we conceded."Looking at all the openings we fashioned and the way we managed the game, I must say it was a good performance in terms of the chances we had. "We didn't really get the rub of the green but that's our own fault. We must be more clinical.”
- Eusebio Di Francesco
The Giallorossi have two games left to finish 2017 - both of them in Serie A.
On Saturday they travel to Turin to face Juventus in an important game at the top of Serie A, before the following weekend they are back at the Olimpico to host Sassuolo.
Goalscorers: Schick (85'); De Silvestri (39'), Edera (72')
Roma: Skorupski; Bruno Peres, Moreno, Juan Jesus, Emerson Palmieri; Gerson (Pellegrini), Gonalons, Strootman (Dzeko); El Shaarawy, Schick, Cengiz (Perotti).
Torino: Milinkovic-Savic; De Silvestri, Lyanco, Moretti, Molinaro; Acquah, Valdifiori, Obi; Edera, Belotti, Niang.
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