The Giallorossi made tough work of the Serie B side, especially in a first half where Luciano Spalletti's side rarely threatened to get out of first gear.
Substitute Edin Dzeko then scored midway through the second half to finally break the deadlock, only for a defensive mix-up to allow Cesena's Garritano to equalise not long after.
Roma then toiled away in search of a winner - with a clear-cut chance only arriving in the very last minute, after Kevin Strootman was adjudged to have been fouled in the box.
Totti - who else? - took on the responsibility, beating Agliardi low to the goalkeeper's right and sending his side into the next round.
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The Giallorossi will now play Lazio in a two-legged semi-final, with the first leg taking place at the start of March.
Needless to say, they will have to step up their performance for those two matches - although the derby atmosphere will surely guarantee that is the case.
While the fans on Wednesday night got their Hollywood ending, in truth the players will not look on this game with too much fondness, realising that perhaps - without taking anything away from the visitors' dogged display - they should have made the evening much easier than it turned out to be.
Nevertheless, against a resolute opponent whose performance belied their current standing in Serie B, progressing into the next round was the main thing.
Coach Luciano Spalletti once again opted to start Mario Rui and Alisson in the cup, with Francesco Totti captaining the side and leading the line.
Elsewhere Leandro Paredes came into midfield to give Radja Nainggolan a rest, with Juan Jesus lining up in defence alongside Kostas Manolas and Antonio Rudiger.
15' - Manolas flashes just wide with a volley on the turn from the edge of the box. The home side controlling the ball.
21' - First substitution. Perotti pulls up with an injury problem, and has to be withdrawn. Dzeko replaces him.
25' - Let off for Roma! A deep free-kick carries all the way to Kone, who thunders a shot off the upright and out.
26' - CLOSE! Totti curls in a brilliant low cross for Dzeko, whose clever shot is just fractionally high and wide!
29' - Alisson makes an acrobatic leap to palm away Rodriguez's powerful header. Good save from the Brazilian.
39' - Another great stop from Alisson, who gets enough on Rodriguez's low strike to keep it out. A let-off for Roma.
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60' - SO CLOSE DZEKO! The striker opens up the space for himself, but his curling shot is fractionally wide!
64' - TWO GREAT CHANCES FOR DZEKO! First Agliardi stops him from point blank range, then he prods Totti's cross wide!
67' - GOAL! ROMA LEAD! Finally, a slick passing move carves Cesena open! Radja puts in El Shaarawy who squares for Dzeko ... and he makes no mistake! 1-0!
74'- Goal - Cesena equalise. A high cross escapes Alisson as he clashes with Manolas and Garritano drives home the loose ball. 1-1.
77' - UNBELIEVABLE SAVE! A short free-kick is absolutely blasted by Totti , but Agliardi pulls off a diving stop!
90' - PENALTY FOR ROMA! Strootman is brought down by the goalkeeper after a slick first touch ... and Roma have the chance to win it!
90' - GOAL! TOTTI! The captain takes on the responsibility ... and drives home a low penalty to win the game for his team. 2-1!
See more match statistics in our match centre.
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"Yeah, I felt a lot of pressure. It was one of the penalty kicks I've felt the pressure most in my career. I hoped not to miss it - I had a lot of responsibility."If the game goes to extra-time anything I don't know what might happen, especially as four or five of us were cramping. The important thing to score that penalty was to be focused, have conviction, and be absolutely sure of what I was doing."
- Francesco Totti
The Giallorossi are back at the Stadio Olimpico in six days' time, as they host Fiorentina in Serie A action on Tuesday night.
The following weekend they travel to Crotone, facing the relegation-threatened side on Sunday 12 February.
Goalscorers: Dzeko (67'), Totti (p. 90'+5); (Garritano 73')
Roma: Alisson; Rudiger, Manolas, Jesus (Nainggolan); Peres, Strootman, Paredes, Rui; Perotti (Dzeko), El Shaarawy (Palmieri); Totti.
Cesena: Agliardi; Rigione, Ligi, Perticone; Balzano, Laribi, Vitale, Kone, Renzetti; Rodriguez, Ciano.
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