Joel Voelkerling Persson and Riccardo Pagano scored the decisive goals at Stadio Tre Fontane to book a last-eight showdown with Juventus.
That game will take place in Turin on 12 January 2022.
"It was a tough game and we knew it was going to be," coach Alberto De Rossi said afterwards.
"My opposite number at Cosenza put them out in an organised, compact formation that didn't give us much space to work in.
"We didn't give away anything ourselves though - the lads did well in working out how to manage the game. Grinding out the victory in the end shows the side are gaining a bit of maturity too."
See photos from the game in the gallery below!
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