Fourth seed Gauff admitted to making a nervous start in the first match on Rod Laver Arena but soon hit her stride to rattle off the last nine games for a convincing 6-3, 6-0 win over Anna Schmiedlova.
"I feel like a lot of my shots that I'm confident in, when you're nervous there is always a second-guessing feeling," said the American teenager.
"In the second set, I was playing much deeper. She also made more errors, too, but I think I was just playing much deeper and much freer. I think I just freed up and everything started to fall in place."
Vondrousova was the first unseeded player to win the women's title at Wimbledon last year but found the shoe on the other foot on Monday as she succumbed to a 6-1, 6-2 upset loss at the hands of Yastremska.
The 2024 crop of qualifiers have caused a host of problems for top players on the first two days of the tournament, not least Dino Prizmic's defiance of 10-time champion Novak Djokovic over four hours on Sunday night.
Men's third seed Daniil Medvedev, who decided to skip the warm-up events this year, was perhaps fortunate that his opponent Terence Atmane retired due to cramp during his opening match on a sunbathed Margaret Court Arena.
The French world number 144, who had taken the first set off the 2021 US Open champion before Medvedev levelled up the contest at 1-1, was holding his own in the third when he started suffering severe cramps in his thigh.
Atmane received a time penalty as he struggled get to the service line and another violation for smashing his racket in frustration but played on and even broke Medvedev's service after a courtside massage.
The 22-year-old finally called time on his first Grand Slam main draw match while trailing the Russian 5-7, 6-2, 6-4, 1-0.
"Tough match, he played pretty well," Medvedev said of the qualifier.
"I played badly at the start of the first set. When I started feeling better physically, he started cramping. It was brutal conditions out there."
There were emotional scenes on Court 13 where Italian qualifier Flavio Cobolli won his first Grand Slam main draw match with a 6-4, 3-6, 6-3, 2-6, 7-5 upset of Chilean 18th seed Nicolas Jarry.
Sebastian Korda, the 29th seed, needed five sets to get past Czech qualifier Vit Kopriva 6-1, 6-4, 2-6, 4-6, 6-2, while French lucky loser Hugo Gaston ousted Spain's Roberto Carballes Baena 6-3, 6-2, 3-6, 6-4.
Stan Wawrinka, the only man apart from Djokovic in the field to have won the title, took Italian Adrian Mannarino to five sets before bowing out 6-4, 3-6, 5-7, 6-3, 6-0.
"Even after the loss, I think there is some good opportunity for me to keep pushing, keep playing some good result, and hopefully I can come back next year," said the 38-year-old, who won the title a decade ago.
Andy Murray, who has lost five finals at Melbourne Park, followed his fellow veteran onto Kia Arena but also fell at the first hurdle, losing 6-4, 6-2, 6-2 to Argentine Tomas Martin Etcheverry.
In the women's draw, Australian qualifier Storm Hunter beat former French Open finalist Sara Errani 6-4, 6-3 to claim her first victory in the Melbourne Park main draw at the sixth attempt.
"To qualify and win my first round of main draw here at Aussie Open, it's insane. It's an amazing feeling," the 29-year-old world number 180 said.
"The crowd was incredible. The Aussie fans really got behind me."
There was no hint of an upset in the first match on 1573 Arena as Elina Svitolina, one of the band of mothers playing at this year's tournament, beat local wildcard Taylah Preston 6-2, 6-2.
Ons Jabeur began her latest bid to become the first Arab and African woman to claim a Grand Slam title with a comfortable 6-3 6-1 win over Yuliia Starodubtseva.
The smiling sixth seed - who fans have dubbed Tunisia's "Minister of Happiness" - came within touching distance of Grand Slam glory previously by reaching the last two Wimbledon finals and finishing runner-up to Iga Swiatek at the 2022 US Open.
"It's very special to come back here," said Jabeur, who crashed out in the second round in 2023 amid health issues. "I decided to rest a bit before the Australian Open and play here.
"Last year wasn't so great for me here and I was a little stressed, but the crowd made me feel less stressed. I saw some guys I've known from 2016 cheering for me when I was playing qualifying. They’ve been following me since qualifying.
"Thank you for coming and supporting me for so many years."
