The second seed looked certain to be making a premature exit when she trailed little-known Croatian Fett 1-5, 15-40 in the deciding set only to save the two match points and reel off six games in a row.
After completing her 3-6 6-2 7-5 victory, Wozniacki revealed she used Fett's inexperience - the 21-year-old is ranked 119 and had never won a grand slam match prior to this tournament - to give herself the belief to come through.
The Dane said: "At 5-1, 40-15, I felt like I was one foot out of the tournament. She served a great serve down the T. It was just slightly out. I was kind of lucky.
"Then I felt her tighten up just slightly. I thought to myself, 'Make her win it, don't give it to her'. When I managed to get it to 5-2, I said, 'Okay, I'm still alive. She still has four more balls to win'.
"She's about to beat the number two player in the world. Obviously she's about 100 in the world. That's a big moment for her. I know how it feels whenever you're close to reaching something really big. I'm very proud of the way I came back."
Wozniacki hails from a very sporty family - her mother Anna played volleyball for Poland while her father Piotr was a professional footballer. But it was growing up competing against her brother Patrik, who is also a footballer, that drove Wozniacki's will to win.
She said: "If you lose a game to anyone in my family, you're going to hear about it for at least a week. That definitely helped me.
"Growing up, I never wanted to lose, and the worst was to lose to my brother. Giving him the opportunity to be ahead was not pleasurable. So I think from that point, you just learn and fight. You don't give up, even if it's a board game, tennis, soccer, whatever."
The day's early headlines were written by 15-year-old Marta Kostyuk, who defeated Olivia Rogowska 6-3 7-5 to become the youngest player in more than 20 years to reach the third round of a grand slam. The teenager, who was given a wild card into qualifying after winning the junior title last year, is causing quite a stir back home.
She said: "I took my phone, and I had to eat at the same time. My mum told me to leave the phone and eat properly. Because I didn't have a normal breakfast, that's why I felt quite bad today on court. I didn't still reply to a lot of people."
In the third round, Kostyuk will play her fellow Ukrainian Elina Svitolina, who went into the tournament as the joint title favourite but struggled in a 4-6 6-2 6-1 victory over Katerina Siniakova.
The fourth seed revealed afterwards that she has not been feeling well and was considering pulling out of the tournament ahead of her first-round match.
Speaking about Kostyuk, she said: "The first time I heard about her was when she won here. It's good, it's pretty cool that I'm playing someone from my country. It's going to be very exciting for me and hopefully I can be ready for that match."
French Open champion Jelena Ostapenko also had to battle, beating Duan Ying-ying 6-3 3-6 6-4 while there was a surprise defeat for Venus Williams' conqueror Belinda Bencic, who lost 6-1 6-3 to qualifier Luksika Kumkhum.
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