Defender Fabio Lucioni stunned the crowd at the Stadio Marc'Antonio Bentegodi with a header in the 60th minute, and the Flying Donkeys could not muster a response.
Reggina, currently 18th in Serie B, are the lowest placed team left in the competition and will make their first appearance in the last 16 since 2008.
Cesena also took the lead against top-flight opposition but the east-coast club were ultimately knocked out of the competition as Atalanta rallied to win 3-1.
Denis Tonucci put the recently-relegated side ahead in the 30th minute, only for Facundo Parra to draw the Nerazzurri level four minutes before the break.
Giuseppe De Luca edged Atalanta towards a showdown with Roma in the 62nd minute before Parra completed his brace eight minutes from time to safeguard the result.
Six-time winners Fiorentina found southern minnows Juve Stabia hard to break down but two second-half goals were enough to set up a trip to Udinese.
The Serie B outfit withstood the Viola attack until Haris Seferovic found a way through in the 52nd minute, with Ahmed Hegazy making it 2-0 before Stabia defender Michele Murolo saw red at the end.
A solitary strike from Cristian Pasquato earned Bologna a 1-0 victory over Livorno and a last-16 meeting with holders Napoli, while Siena sealed their progression with a 2-0 win over Torino in the only all-Serie A fixture.
Ze Eduardo nudged the Bianconeri ahead one minute after the break and Reginaldo doubled the advantage in the 77th minute. Siena will play Lazio next at the Stadio Olimpico.
Catania challenge Cittadella for a trip to Parma and Cagliari play Pescara to earn a daunting trip to Juventus, in next week's remaining fourth round ties.
Wednesday's Coppa Italia Results
Atalanta 3-1 Cesena
Chievo 0-1 Reggina
Bologna 1-0 Livorno
Siena 2-0 Torino
Fiorentina 2-0 Juve Stabia