Former world number one Wozniacki was simply outplayed by the 21-year-old Croatian for large parts of the match but won six straight games to avert the upset and claim her place in the third round.
"That was crazy, I don't know how I got back into the match," said a relieved Wozniacki.
"She's a tricky opponent, she had nothing to lose, and I think she realised she was at 5-1 and she let off the speed a little bit.
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"I thought, 'this is my last chance and I'll have to go on the attack'. Then things were going my way and I thought ' this is my chance'."
Fett, ranked 119th in the world, played some brilliant shots throughout the contest and rattled off eight straight points to win the opening set in 33 minutes.
Fired by the first of two rows with umpire Richard Haigh over delayed challenges, Wozniacki worked her way back into the contest to take the second stanza when Fett hit a return long.
The Croatian refused to fade away, though, and Wozniacki threw her racket to the ground in disgust after being broken for 3-1 with Fett driving home her advantage to stand on the brink of the third round at 5-1 40-15 up.
Fett twice served for the match but nerves got the better of her and Wozniacki took full advantage to set up a third-round meeting with Belgian Kiki Bertens or American Nicole Gibbs.
The Dane looked every inch the world number two as she briskly served out for victory, which came when a crestfallen Fett dumped a backhand into the net after a shade over two and a half hours in the sun on Rod Laver Arena.
"Experience was crucial today," said Wozniacki, who is on her 11th visit to Melbourne Park as she continues her quest for a maiden grand slam title.
"I've been out here so many times and I knew how she would feel being out here against me and having the chance to win."
(Reporting by Nick Mulvenney in Sydney, editing by Peter Rutherford/Amlan Chakraborty)
