The former striker claims his side were not quick enough against Inter and did not have the confidence needed to win in their domestic cup tie against Leonardo's men
The Rome club needed a victory at San Siro against Inter after losing the first-leg at home 1-0 but could only muster a 1-1 draw.
Montella's men will end the season without a trophy, with the coach believing beating Inter was always going to be a difficult task.
"We were good in the first-half but lacked a bit of pace and a bit of confidence. We carried on to the end, but we were playing Inter and you can't expect to get man opportunities against them," the coach told Italian media after the game.
Roma officials were disappointed with the amount of added time, believing stoppages in the game deserved more than the allocated three minutes.
However, Montella refused to blame the issue for his side's exit from the competition, despite his team looking the likelier to score in the closing stages of the game.
"I will not speak of added time, I think only about the game. We tried to qualify [for the Coppa Italia final] and this is all I wish to talk about," he added.
The club return to Serie A action at the weekend with a trip to Catania. Victory is vital for fifth-placed Roma as they sit two points behind Udinese, who occupy the final Champions League qualification spot with only two games to go.