Edin Dzeko and Alexis Sanchez scored in either half for the Nerazzurri, who made home advantage count as they moved on to the semi-finals of the competition.
After Dzeko had given the hosts the lead early on the Giallorossi regrouped and eventually started to make headway in the contest, with Nicolo Zaniolo, Jordan Veretout and Tammy Abraham among those to go close.
But Sanchez's impressive strike from distance midway through the second half ultimately decided the contest.
Jose Mourinho's side are back in action on Sunday - they travel to face Sassuolo in Serie A.
INTER MILAN 2-0 ROMA
SCORERS: Dzeko (2'), Sanchez (68')
INTER (3-5-2): Handanovic; D’Ambrosio, Skriniar, Bastoni (De Vrij); Darmian, Barella (Vecino), Brozovic, Vidal (Calhanoglu), Perisic (Dumfries); Dzeko (Lautaro), Sanchez
ROMA (3-5-2): Rui Patricio; Mancini, Smalling, Ibanez (Kumbulla); Karsdorp, Sergio Oliveira (Pellegrini), Veretout (Cristante), Mkhitaryan, Vina (El Shaarawy); Abraham (Afena-Gyan), Zaniolo
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