The hosts, who drew 1-1 at home with Inter in the league on Sunday, raced into a 2-0 lead after Alessandro Florenzi rose highest to head in after 13 minutes and striker Mattia Destro also nodded in a cross just after the half hour.
Rodrigo Palacio pulled one back for Inter a minute before the break when he escaped the offside trap and turned in Esteban Cambiasso's free-kick with the home defence all at sea.
Inter are fourth in Serie A and Roma seventh but Zdenek Zeman's side created the better chances and just about deserved to take a lead into the second leg as they target a cup triumph to guarantee access to next season's Europa League.
The second leg will not be played until April with the long wait due to the generally unloved Coppa Italia's lack of clout in fixture programming.
Juventus drew 1-1 at home with Lazio in Tuesday's semi-final first leg with their second leg taking place next Tuesday.