Lecce, who were in Serie A as recently as 2012, looked to have forced extra-time in their clash at Spezia, but the third-tier side were unable to hold on.
Spezia edged the encounter in general play, but their fans were made to wait as they desperately sought a late goal to secure progression.
Eventually, the winner did arrive as defender Filippo De Col grabbed the all-important goal to earn a 1-0 victory for Nenad Bjelica's men.
Benevento twice found themselves a goal behind to hosts Virtus Entella in Chiavari, but the third-tier side did manage to avoid a regulation-time defeat.
The visitors were staring cup elimination in the face, only for Alessandro Marotta to net in the third minute of stoppage time, but they eventually lost 7-6 on penalties.
There was to be no such drama for Cittadella, who defeated Pontedera 2-1 at Stadio Pier Cesare Tombolato.
Pontedera's Saverio Madrigali cancelled out Federico Gerardi's opener just before the break, but Cittadella's Alessandro Sgrigna scored the decisive goal 10 minutes into the second period.
Modena edged Monza 3-2 on penalties, after normal and extra time finished 0-0.
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