Lucic-Baroni, who had taken Simona Halep out of two of the last five Grand Slams, came out swinging, powering through the first set in just 31 minutes, 6-3, and, after a long, grueling second set, coming within four points of victory as the two players were locked in at 3-all in the second set tie-break.
But Wozniacki rattled off four points in a row from there to clinch the second set and took complete control of the third set, building leads of 4-0 and 5-1. Lucic-Baroni caught fire again to close that gap to 5-4 but the No.2-seeded Dane lifted her level one last time to close the match out, 3-6, 7-6(3), 6-4.
The match was so close that Lucic-Baroni actually won more points than Wozniacki, 107 to 104.
"She's been playing really well, and she usually plays well against the top-ranked players - she's beaten Halep two times in the last year - so I knew this wasn't going to be easy," Wozniacki said.
"I just tried to take it one ball at a time and stay positive. It was very close. Sometimes it goes one way and sometimes the other - I'm just happy with this win and happy to be through to the next round."
The only other first round match on the schedule saw another favorite, Roberta Vinci, take the court, but she wasn't able to escape the upset - the No.3-seeded Italian put forth a valiant fightback from 6-1, 4-2 down to push it to a third set but fell to Serbian qualifier Aleksandra Krunic anyway, 6-1, 6-7(3), 6-1.
"It wasn't my day today," Vinci said. "I tried to be aggressive and tried to play more to her backhand, but she played well. I don't usually like courts this slow - I prefer to play on hardcourts that are faster. But I tried my best. She was just too good. I have to think about the next tournament now."
The first three second round matches of the week also took place Wednesday, and there was another upset, as Margarita Gasparyan and her gorgeous one-handed backhand won an all-WTA Rising Star clash against the No.6 seed, Camila Giorgi, 6-3, 2-6, 6-3. Giorgi was a finalist in Linz a year ago.