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The striker slotted home with just under 20 minutes remaining to finally break the deadlock against a resolute Sampdoria side, on a day where heavy rain made life difficult for players at the Stadio Olimpico.
The Giallorossi dominated possession for large swathes of the game but struggled to create many clear-cut openings, although Emil Audero was called upon on a number of occasions to make important stops - least of all from Dzeko, moments before he finally did score.
Chris Smalling also hit the crossbar as Lorenzo Pellegrini threatened multiple times - but in the end it was Dzeko who made the difference, steering home expertly from Rick Karsdorp's low cross to finally find the back of the net.
That goal sparked the visitors into a more attacking setup, but Roma held strong defensively in the final 15 minutes to see out the game and earn a first three points of the new calendar year.
Paulo Fonseca's men are back in Serie A action on Wednesday, as the side visit Crotone.
ROMA 1-0 SAMPDORIA
GOALSCORER: Dzeko (72')
ROMA (3-4-2-1): Pau Lopez; Mancini, Smalling, Ibanez; Karsdorp, Veretout, Villar (Cristante), Peres; Mkhitaryan, Pellegrini; (Perez) Dzeko (Mayoral)
Unused subs: Farelli, Boer, Jesus, Kumbulla, Diawara, Darboe, Providence, Pogdoreanu
SAMPDORIA (4-4-1-1): Audero; Yoshida (Leris), Tonelli, Colley, Augello; Candreva, Thorsby, Ekdal, Jankto; Verre (Damsgaard); Quagliarella (La Gumina)
Unused subs: Ravaglia, Letica, Rocha, Silva, Ramirez, Askildsen, Regini, Bereszynski
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