Gauff faced match points in the second set against the talented American teenager, but she managed to fight back to win the match with a decider.
The match eventually ended 5-7, 7-5, 6-2 as Gauff managed to just about hold on to her status at the Rome tournament, following Gauff's difficult match against Solana Sierra too.
Speaking in her post-match interview, Gauff revealed: "I think I'm just proud. Iva played really good tennis and the conditions today weren't easy. We had win, we had airplanes, we had phones ringing. Everything that was going on, but yeah I'm really happy."
Gauff said a mid-match team talk from her team helped her turn the match around against Jovic, when she looked on the cusp of exiting the competition.
"Thanks to my team, that pep talk definitely helped me mentally," she said.
In a later interview with Sky Sports' Gigi Salmon, Gauff added: ""It was tough to find my rhythm. It was really windy today, I felt like I was missing a lot of balls. I'm really happy with my mentality. I wasn't playing my best in some moments, but I was able to find a way.
"I learned a lot over the years when it comes to those tough moments. It's just trying to relax and play tennis, but it's easier said than done.
"Obviously my previous round was mentally not good enough and today was a complete 180, so I think that shows I'm able to be positive for the most part and I need to continue to do that."
Gauff will now play either Mirra Andreeva or Elise Mertens in the quarter-finals. The American's draw has opened wide open since Aryna Sabalenka's surprise defeat to Sorana Cirstea.
If Gauff reaches the final four of the Italian Open she will play the winner of Cirstea vs Jelena Ostapenko, the latter of which beat Anna Kalinskaya to reach the quarterfinals.
Gauff is yet to win the Italian Open in her career, but she did reach the final in Rome last year. The American was defeated by Jasmine Paolini in the final.
The star has an excellent record at the Italian Open, however, having reached the semi-finals or later on three occasions throughout her career.
She holds a 76% win rate in Rome, having won 20 of her 26 matches so far at the Italian Open. With the draw opening up, there's every chance Gauff can go one further this year and lift the trophy.
It would be the perfect preparation for Roland Garros, where the American is looking to defend the title she won last year in Paris.
Elsewhere, seventh Elina Svitolina defeated Nikola Bartunkova 6-2, 6-3.