The teenager prevailed 6-3, 6-4 over a below-par Osaka, who made 30 unforced errors during the 67-minute contest.
Gauff survived a brief rally from Osaka early in the second set, when the Japanese player's break back looked to be a turning point.
The world No. 67 remained impressively focused and served out another landmark victory at the first time of asking to reach the last 16 in Melbourne.
Gauff will face either world No. 15 Sofia Kenin or China's Zhang Shuai in the next round.
In the wake of her victory, Gauff said:
"Oh my gosh, I don't even know [where that came from] … honestly, what is my life? Like, oh my gosh. Two years ago I lost first round in juniors. And now I'm here. It's crazy. I'm on Rod Laver Arena. I can't believe this."
The opening set of the contest was a high-quality affair. Both players started aggressively and held their opening service games confidently.
The first break-point opportunity did not come until the eighth game, when Gauff was 4-3 ahead.
At 30-40 behind, Osaka netted a backhand, a worrying sign of things to come for the 2019 champion, and Gauff took the first of her three set points in the next game with a huge serve:
Osaka's troubles continued at the start of the second set as she made further uncharacteristic errors.
A poor miss long and wide was indicative of her struggles and handed Gauff a break and a 1-0 lead she had barely had to work for.
However, in the next game, Osaka improved considerably, winning two excellent points at the net to put pressure on Gauff's serve.
And the American gave back her break immediately after double-faulting:
It looked like a potential turning point as Osaka had broken out of her malaise—she held to love for 2-1—and Gauff was looking nervous for the first time in the match.
However, Gauff steadied herself with a couple of strong holds of serve, and then she took full advantage of another error-strewn game from Osaka to break again and go 4-3 ahead.
At 5-3, the 15-year-old nearly broke to win the match, but Osaka came up with a couple of huge winners, usually a regular feature her game, to extend the contest.
It only amounted to a stay of execution for Osaka, though, as Gauff held to love to seal another huge victory in her fledgling career.