The result bodes well for the American, who has won every tournament in which she has beaten Muchova, including the 2023 US Open.
"I think today I didn't panic. I felt like the second set could have easily went my way," Gauff said. "So I knew I just had to capitalise on those chances in the third set and I did that."
What started as a demolition job quickly turned into a compelling see-saw battle at Melbourne Park, with Gauff storming to a blistering 5-0 lead in just 20 minutes.
Muchova, who arrived with two top-10 wins under her belt in Brisbane, did not get on the board until the sixth game and by then the opening set had already slipped away as Gauff closed it with clinical efficiency.
The tide turned in the second set, however, with Muchova breaking twice to establish a 4-1 lead as Gauff struggled with her serve and made double faults.
When the chance to serve for the set arrived Muchova did not waste it, serving to love to claim a set over Gauff for the first time in five meetings.
Gauff, who constantly looked to the players' box in the second set, came out swinging in the decider, forcing more errors from Muchova to go up 4-1.
A thrilling seventh game had both players scrambling in long rallies as the momentum shifted back and forth until Gauff fired a backhand winner and screamed in triumph.
Muchova saved three match points before Gauff sealed progress to the last eight.
"She plays with so much variety," Gauff said. "You never know what you're going to do.
"In that last game I was trusting my second serve, which I feel like is a stepping stone in the right direction to be able to hit as many second serves in the court that I did, especially in the two games that were long holds."