The 27-year-old Pole, ranked world No. 9 by the Association of Tennis Professionals (ATP), defeated Baez 5-7, 7-6, 6-4 in the round of 16 on Tuesday.
The seventh-seeded Polish player will now take on 14th-seeded American Tommy Paul for a place in the semifinals.
Hurkacz is playing for his ninth career singles title in Rome.
The clay-court tournament in the Italian capital runs until Sunday.
(gs)
Source: IAR, polskieradio24.pl, PAP