Edinson Cavani gave the hosts a 12th-minute lead but Cristian Pasquato on 38 minutes and Panagiotis Kone in stoppage time helped Bologna set up a last-eight clash with Inter Milan, who beat second-tier Hellas Verona 2-0 on Tuesday.
Greek midfielder Kone had also scored late on for modest Bologna when they won 3-2 at Napoli in the league last weekend.
Lazio also won through after a 4-1 penalty shootout success at home to Siena following a 1-1 draw. The match was played in a mostly empty stadium in the Italian midweek afternoon such is the lack of interest from fans in the competition.
They will face Catania in the January quarter-finals after the Sicilians progressed last week along with AC Milan, Juventus and AS Roma, who will host Fiorentina following the Florence side's 1-0 win at Udinese on Wednesday.
Borja Valero netted for Fiorentina in the 36th minute.
The standout tie of the pre-drawn last eight is Serie A champions and last season's beaten finalists Juventus travelling to Milan.
Serie A title-chasers Napoli were docked two points on Tuesday in Italy's latest match-fixing shame as three players were banned for between six months and three years over bets placed on a match with Sampdoria in 2010.
Paolo Cannavaro, brother of Italy great Fabio, and fellow Napoli defender Gianluca Grava received six-month bans for not reporting evidence of match-fixing while former goalkeeper Matteo Gianello was banned for three years and three months for orchestrating the fix.
Napoli, who said the club had nothing to do with the issue, have appealed.
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