Pegula posted a 6-3, 6-4 victory over 24th-ranked Qinwen Zheng of China in 1 hour, 38 minutes, while Gauff fired six aces en route to a 6-3, 6-2 win over Bernarda Pera in 61 minutes.
Pegula and Gauff could meet in the quarterfinals, provided they win their second-round matches on Wednesday against Colombian qualifier Camila Osorio and lucky loser Jodie Burrage of Great Britain, respectively.
Also on Tuesday, Frenchwoman Caroline Garcia secured a 6-1, 7-5 win over Madison Brengle and fourth-seeded Ons Jabeur of Tunisia captured a 6-3, 6-2 victory over Italian Jasmine Paolini.
Eighth-seeded Beatriz Haddad Maia had a much tougher time of it before notching a 3-6, 6-3, 7-6 (3) win over Marie Bouzkova of the Czech Republic.
Bad Homburg Open
Second-seeded Liudmila Samsonova outlasted Linda Noskova of the Czech Republic to punch her ticket to the quarterfinals.
Samsonova recorded 18 aces to secure a 6-4, 7-6 (4), 6-3 win over the 18-year-old Noskova.
Fifth-seeded Bianca Andreescu was not as fortunate. Spain's Rebeka Masarova registered seven aces en route to a comprehensive 6-3, 6-2 victory over the Canadian, a former U.S. Open champion and the runner-up at this tournament last year.
Also on Tuesday, eighth-seeded Varvara Gracheva notched a 6-2, 7-5 win over Italian Sara Errani.
