Giovinco, who has started only two Serie A games this season, opened the floodgates when he curled in a free kick after five minutes.
Carlos Tevez, Fernando Llorente, Andrea Pirlo and Giuseppe Buffon were among the Juve regulars who were rested, while Arturo Vidal was ruled out with tonsillitis.
Roberto Pereyra scored his first goal for Juventus with a neat chipped finish over Verona goalkeeper Rafael and Giovinco gave the hosts a 3-0 halftime lead when he turned in Stephan Lichtsteiner's pass from close range.
Paul Pogba fired the fourth early in the second half, Alvaro Morata converted a penalty and Kingsley Coman completed the rout with his first goal for the club as Juve earned a quarter-final tie away to Parma.
Nene provided the only reply for Verona, who will not be looking forward to returning to the Juventus stadium for a league match against the same opponents on Sunday.
(Writing by Brian Homewood in Berne; editing by Toby Davis)
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