Diego Perotti converted from the penalty spot after just 10 minutes to give the Giallorossi the early lead many perhaps anticipated - but the further goals many may have been expecting did not materialise, as the visitors defended doggedly for the remaining 80 minutes.
Edin Dzeko had a number of half-chances foiled as both he and Aleksandar Kolarov hit the woodwork, with Eusebio Di Francesco's side eventually forced to settle for the narrowest of winning margins.
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Nevertheless, the result keeps the squad moving in the right direction - with Saturday's home game against Bologna giving them an almost immediate opportunity to deliver a sharper performance.
Perotti, with his first goal of the campaign, was the hero on the night. He converted with aplomb after Kolarov had been fouled in the box, with the referee awarding the spot-kick with no hesitation.
The result moves Roma to 21 points from nine games, fifth in the standings.
Coach Di Francesco made six changes from the side that won at Torino on Sunday afternoon, with Rick Karsdorp getting his first start for the club in front of the home fans.
Cengiz Under and Gerson also both cacme into the side, while Federico Fazio partnered Hector Moreno in central defence - with Maxime Gonalons in midfield and Radja Nainggolan captaining the side.
9' - PENALTY! Kolarov goes down in the box after a challenge from Sampirisi ... and the referee points straight away!
10' - GOAL! No mistake from the spot... Perotti with his first goal of the season! 1-0!
18' - POST! Was that a shot?! Kolarov's inswinging delivery from out wide on the left very nearly fools Cordaz!
36' - Dzeko wrestles his way into a dangerous position inside the box, but the ball goes narrowly wide.
37' - Chance for Crotone. Rohden does well to get space inside the box but his shot is comfortable for Alisson.
40' - Kolarov's low cross is met by Dzeko, but the defender gets a touch to it and Cordaz is able to save.
45' - Big chance on the stroke of half-time! Perotti's cross finds Dzeko but his header loops wide of the far post!
See more images in our dedicated match gallery.
53' - POST! A bursting run from Cengiz eventually sees the ball fall to Dzeko ... who smashes a drive off the woodwork!
54' - Karsdorp follows up with a snapshot drive from 25 yards that stings Cordaz's palms.
56' - Another Kolarov-Dzeko combination ... another Cordaz save. Frustrating for Edin!
78' - Defrel curls narrowly wide from the edge of the box.
90' - The ball falls to Simic from a Crotone corner but Alisson saves and holds.
90' - Dzeko turns and shoots but he scuffs it slightly and Cordaz is able to hold.
See more game statistics in our dedicated match centre.
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“We won the game, but we could have done better in an offensive sense, taking the chances we created. We hit the post twice – perhaps it’s a bit of bad luck because it’s happened numerous times this year. But we could have been more clinical in front of goal, especially having created so much. I made changes and we still got three points – that’s important as we chase the teams ahead of us.”
- Eusebio Di Francesco
“It’s an amazing feeling, I feel like a little kid again with the joy of playing out on the street with a ball. I’ve been training with the squad for two weeks now and I’m happy to get 80 minutes under my belt for my first appearance for this great club.“Obviously I’m also very happy with the win and the three points.”
- Rick Karsdorp
Roma have two more home games ahead of them now, as they return to the Olimpico on Saturday to host Bologna - the team Di Francesco's son, Federico, plays for.
Chelsea then come to town the following Tuesday for a crucial Champions League encounter for both sides.
Goalscorers: Perotti (10' (p))
Roma: Alisson; Karsdorp (Florenzi), Fazio, Moreno, Kolarov; Gonalons, Nainggolan, Gerson (Pellegrini); Cengiz (Defrel), Dzeko, Perotti.
Crotone: Cordaz; Sampirisi, Ceccherini, Ajeti, Pavlovic; Izco, Mandragora, Barberis, Stoian; Nalini, Budimir.
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