Osaka, the four-time Grand Slam singles champion, won at Rod Laver Arena and did so by taking the first five games of the match, before Osorio finally scratched one out herself.
Osaka will take on Madison Brengle -- the American player beat Dayana Yastremska of Ukraine 6-1, 0-6, 5-0, ret. -- in the second round as she looks to build some momentum for what could be a lengthy tournament run.
The former No. 1 player in the world, Osaka slid down the rankings last year after taking time off following her withdrawal at the French Open. She is seeded 13th at the Australian Open, where she won in 2019 and 2021.
"It always feels special to come back here," she said after Sunday's victory.
The early success seems promising for Osaka, as Osorio is ranked 50th in the world but was making her main draw debut at Melbourne Park.
"I thought she played amazing," Osaka said of her opponent. "Overall, I'm just happy to be here, I'm happy to see everybody in the audience, and I hope we gave you a great performance."
After winning here last year, capturing her second Australian Open title in three years, Osaka pulled out of the 2021 French Open before the second round then sat out Wimbledon. She played at the Tokyo Olympics, where she lit the cauldron, but ended her 2021 season early after a third-round loss and a teary news conference at the US Open.
Two of her goals for 2022, she said last week, were to stay completely composed on the court and off, and to enjoy the game.
The first test of Osaka's new approach to tennis might have been when she completely whiffed on an overhead to give her opponent a break point. Osaka didn't chuck her racket. She didn't roll her eyes. She smiled.
"There are situations where I previously would get upset. But at this point in my life, like, I'm here because I want to be here and because I find that it's fun for me,'' Osaka said. "Might as well enjoy it while I still can.''
It was Wang's first win on tour since last year's French Open, where she beat Su-Wei Hsieh in the first round before losing to Gauff in the second.
Gauff saved four match points from 5-0 down on Monday, but that was the only good news in a match in which the American made 38 unforced errors.
"You know I lost two times against her last year, so I just tried my best to focus on court,'' Wang said.
Sakkari felt the pressure of being the first to move on but said the experience was "weird ... in a good way."
"I was a little bit stressed, opening the tournament on this wonderful court," she said. "It's a privilege to have my first match here."
French Open champion and fourth-seeded Barbora Krejcikova advanced with a 6-2, 6-0 win over Andrea Petkovic. Krejcikova lost to Paula Badosa in three sets at the Sydney WTA tuneup tournament final on Saturday.
Tokyo Olympics gold medalist Belinda Bencic followed by defeating Kristina Mladenovic 6-4, 6-3, moving into a second-round match against 20-year-old American player Amanda Anisimova, who beat qualifier Arianne Hartono 2-6, 6-4, 6-3. And 15th-seeded Elina Svitolina registered a 6-1, 7-6 (4) win over Fiona Ferro.
Also, a back injury forced ninth-seeded Ons Jabeur to withdraw before her scheduled first-round singles match. The injury forced the Tunisian player, who reached the third round at Melbourne Park last year, to retire from her quarterfinal at a Sydney lead-up event last week.
She was replaced in the main draw by Romania's Irina Bara, a lucky loser from the qualifying tournament. China's Saisai Zheng was another late withdrawal on the Open's first day. She was replaced by Nao Hibino of Japan.
The Associated Press contributed to this report.