Despite going behind early on to Arturo Calabresi's header, Jose Mourinho's side picked themselves up and eventually turned the tie around - with Marash Kumbulla, Tammy Abraham and substitute Eldor Shomurodov all getting themselves on the scoresheet.
The Giallorossi will now face Inter for a place in the semi-finals, after they edged past Empoli in their last-16 tie on Wednesday.
An admittedly underwhelming first half performance from the hosts was at least salvaged by Kumbulla's equaliser, as he rose at the back post to nod in Abraham's flicked-on corner in a fashion not dissimilar to Lecce's opener.
Mourinho then changed his side around in the second half, with the introduction of Nicolo Zaniolo and Henrikh Mkhitaryan in particular adding a creativity and threat that had been lacking in the first half.
But it was a striker's turn and finish from Abraham that ultimately turned the game on its head, with Mario Gargiulo's red card moments after effectively erasing any hopes Lecce may have had of staging their own comeback.
When Shomurodov then breezed through on goal and finished expertly at the near post, the game was wrapped up in Roma's favour.
The side are next in action on Sunday - they travel to face Empoli in Serie A in the final appointment before the international break.
ROMA 3-1 LECCE
SCORERS: Kumbulla (40'), Abraham (54'), Shomurodov (81'); Calabresi (14')
ROMA (4-2-3-1): Rui Patricio; Karsdorp, Ibanez, Kumbulla, Maitland-Niles (Vina); Cristante, Veretout (Mkhitaryan); Perez (Zaniolo), Sergio Oliveira, Afena-Gyan (Shomurodov); Abraham (Zalewski)
LECCE: (4-3-3): Bleve (Gabriel); Gendrey, Calabresi, Dermaku (Lucioni), Barreca; Gargiulo, Blin, Helgason (Bjorkengren); Di Mariano (Coda), Olivieri, Listkowski (Faragò)
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