Gustavo Varela scored twice at the Brianteo, opening the scoring after a deflection midway through the first half before adding a simple tap-in from a Patrick Cutrone assist right before the break.
Cutrone turned provider again early in the second half, threading a pass for Nathan Forson to finish first time for the third goal, sealing Monza's passage to the next round, where they will face the winner of Torino and Carrarese.
Speaking to Monza-News after the match, Juric praised several of his younger and newer players.
"I'm very pleased with Cutrone because he always gives his all," the Croatian coach said. "Forson has great potential and a real desire to improve, and it's the same with the other young players — working hard with them, I know we can make them better. Our lads have real quality."
Juric also addressed the fitness and transfer situation facing his squad ahead of Monza's Serie A opener away at champions Inter on August 22.
"Pessina got injured and won't be available for a while — we're short in midfield, so we need to assess things carefully. There's still a lot to do in the transfer market, we need experience."
He gave an update on Andrea Petagna's situation, with the club reportedly assessing his future given financial constraints.
"We're weighing things up with Petagna. The club wants to make a bit of a revolution, and we don't have huge financial resources. For me Petagna is a great lad, and he played very well today — whether he stays or goes, he's going to have a great season regardless."
Pessina's absence leaves Juric light in central midfield for the visit to Inter, with the captain expected to undergo surgery on his knee injury and face a lengthy spell on the sidelines. The head coach's newly patched-together midfield, built around Foe Ondoa and academy graduate Mathis Mout — who made his professional debut in the win — will need reinforcement before the transfer window closes if Monza are to cope with the early-season fixture list.
Monza's next Coppa Italia assignment, in September, will be against the winner of Torino and Carrarese, but the club's immediate focus turns to the Serie A opener at San Siro against holders Inter.