Fourth seed Gauff said ahead of the opening Grand Slam of the year at Melbourne Park that she wanted to win "multiple" major titles after her breakthrough at Flushing Meadows.
The 19-year-old put the plan into motion with a 6-3, 6-0 rout of Slovakia's 68th-ranked Anna Karolina Schmiedlova on a baking-hot Rod Laver Arena.
"I was a little nervous in the first set and I think I did well returning and then I found my serve towards the end (of the set)," said Gauff, who successfully defended her Auckland title in the lead-up.
Her reward is a clash with unseeded American Caroline Dolehide in the second round.
While she lived to fight another day, it was curtains for Vondrousova, who slumped 6-1, 6-2 to Ukrainian qualifier Dayana Yastremska.
The Czech seventh seed pulled out of the warm-up Adelaide International last week with a hip injury and said she was also struggling with a sore shoulder, which affected her serve.
"I didn't practise much before the tournament because I was injured," she said after becoming the highest seed to depart so far.
"She played a good match, I didn't play good, that's why (I lost)."
World number 93 Yastremska, whose best showing at Melbourne Park is a third-round appearance in 2019, sported the slogan "Be brave like Ukraine" on her bag and said she wanted to keep the war in her homeland in the spotlight.
"I'm very proud of Ukraine, proud of the people, proud of the warriors and the civilians," she said, after setting up a clash with unseeded Frenchwoman Varvara Gracheva.
Another Ukrainian, Elina Svitolina, also progressed, demolishing Australian Taylah Preston for the loss of just four games.
Nineteenth seed Svitolina, a two-time quarter-finalist in Melbourne, will face unseeded Viktoriya Tomova next after the Bulgarian eased past American Kayla Day.
- Positive mindset -
New mum Osaka, 26, is the main attraction on centre court in the evening session, having stepped away from the sport in September 2022, citing mental health concerns.
After giving birth to a baby girl, Shai, in July, the Japanese star decided to return for the 2024 season.
But she faces a huge challenge first up against 16th-seeded Frenchwoman Caroline Garcia.
"I have a much more positive mindset and a much more grateful mindset," four-time Grand Slam champion Osaka said of returning to Melbourne, the scene of her title-winning exploits in 2019 and 2021.
They will follow on from local hope and 10th seed Alex de Minaur's showdown with big-serving Canadian veteran Milos Raonic.
In other men's action, seventh seed Stefanos Tsitsipas meets face Belgian lucky loser Zizou Bergs.
Tsitsipas, who powered into the final last year but was no match for a dominant Novak Djokovic, was due to meet Italian Matteo Berrettini. But the former world number six withdrew with a foot injury on Sunday.
In another standout match, dangerous Canadian Felix Auger-Aliassime will spar with Austrian 2020 US Open champion Dominic Thiem.
The winner could face world number three Daniil Medvedev in the third round, should the Russian get past French qualifier Terence Atmane.
Medvedev, a Melbourne finalist in 2021 and 2022, has pledged to be "more mature" this year and cut out the tantrums that are a hallmark of his game.
Other players in action include five-time Australian Open finalist Andy Murray and women's sixth seed Ons Jabeur.