Gauff, seeded fourth this year, beat the 14th-seeded Jovic 6-1, 6-4 in 80 minutes.
The match marked a rare instance in which two-time Grand Slam champion Gauff, still only 22, faced a younger American in the 18-year-old Jovic, who pushed Gauff in the second set after being routed in the first.
"She plays beyond her years," Gauff said. "I'm looking to many more battles out here."
Gauff said she served "really well" and called the win, along with a second-round victory over Spain's Paula Badosa, her best matches so far this year.
Broken to open the contest, Gauff reeled off six straight games to take the first set.
Gauff said some of the marathon rallies left her breathless, but she came out on top of them more often than not.
Jovic picked up the pace in the second set, rushing the net behind strong shots to keep Gauff on the defensive.
Their net exchanges won loud applause from fans, and Gauff finished the match with a surprise serve-and-volley ambush, landing a volley that Jovic couldn't put back in play.
Gauff will next face French Open champion and fifth-seeded Mirra Andreeva of Russia in the quarter-finals. Gauff has won all five of their prior meetings.
