The US Open champion, who has made at least the semi-finals of the past three Grand Slams, broke five times in the match to equal her best run at the All England Club.
The American world number two did not have it all her own way in the first set against the world 298.
But ultimately she had too much in her locker for her unknown opponent, firing 16 winners to Kartal's six and racing through the second set without dropping a game in 22 minutes.
"This is my first time playing a British player here so I was little nervous because I knew you'd be for her, which is understandable," said Gauff.
"Being an American I know how it feels when I'm at the US Open. When I play at home it gives me an edge. Thankfully you guys (the crowd) were pretty nice to me so that helped."
Gauff, 20, made her career breakthrough at Wimbledon as a qualifier in 2019, reaching the fourth round, and made the same stage in 2021.
She faces compatriot Emma Navarro in the last 16.
