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The Giallorossi will face Juventus in the last-eight of the competition, with the game taking place in Turin next Wednesday.
On a cold night at the Stadio Tardini, the Giallorossi were made to work for their victory - with their overall dominance of proceedings finally rewarded in the second half.
Pellegrini broke the deadlock with a fine finish following a smart one-two with Nikola Kalinic, before killing off the game from the penalty spot with 20 minutes remaining.
The game even afforded coach Paulo Fonseca a chance to experiment, as he opted for something of a 3-4-2-1 shape that saw Bryan Cristante drop into a central defensive role and the wing-backs push even higher up.
Kalinic got 90 minutes under his belt, while Bruno Peres returned to action in a Roma shirt as a second half substitute.
The Giallorossi are back in action this weekend in Serie A, as they travel to face Genoa on Sunday evening.
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