The Serbian player will face Lorenzo Musetti of Italy in the semifinals after he emphatically beat Daniel Elahi Galan 6-3, 6-0.
The other semifinal pits Berrettini against Mackenzie McDonald of the United States.
Berrettini cruised to a dominant 6-2, 6-3 victory over Taro Daniel, much to the delight of the passionate home crowd, which also included his parents.
“It’s super important and I’m super proud that they’re here,” Berrettini said.
“Obviously without them I wouldn’t be here. . . . When I started playing tennis, they were the ones that brought me everywhere, so I’m so happy that they’re here, that they can enjoy this atmosphere, and hopefully they’re proud of me.”
McDonald rallied to beat Chinese qualifier Zhang Zhizhen 4-6, 6-3, 6-2.
Naples was awarded a single-year license to host the ATP 250-level tournament.