The world number three took one hour and 28 minutes to overcome her German opponent 6-3 6-4.
The American, a two-time Grand Slam winner, broke Lys three times in the opening set, converting three of her four break point opportunities.
“I’m happy with how I played – she’s a tough opponent and she hit some incredible shots on the run but I tried my best to play aggressive and good tennis,” said Gauff.
“I think staying confident in my game [was important]. Not being too passive when I had the lead, I played one passive point in the last game but after that I played good tennis.”
Gauff, who has struggled with her serve at times this season, had an impressive 79% first serve percentage throughout the match.
The 21-year-old, who was knocked out of the opening round of Wimbledon after winning the French Open earlier this year, is now the highest-ranked player left in the competition.
She will face either Amanda Anisimova or Jasmine Paolini in the final four