The US Open winner sent over nearly two dozen winners and overcame issues with her serve, including six double faults, to set up a meeting with the winner of a match between China's Wang Yafan and Ukrainian Dayana Yastremska.
"I'm very happy with how I played. Conditions (were) very slow and muggy but I got through it," said Gauff.
The finalist in 2022 showed early signs of nerves as the unseeded Zidansek broke her serve in the opening game of the match.
But Gauff broke back in the sixth and eighth games, getting the decisive second-set break in the ninth game before pumping her fist with satisfaction as Zidansek sent the ball out of bounds on match point.
The American has revamped her serve since picking up her maiden major on home soil last summer but has not been able to snap a three-year clay-court drought before arriving at Roland Garros.
"I felt like I could have, especially in the second set, probably won it a little bit sooner, maybe 6-3 6-2," Gauff told reporters.
"But with the matches, I mean, there's always things you can do better, and I try not to over-analyse everything."