Badosa won nine straight points while Gauff struggled with her serve, and led 2-0 in the second set before cruising to a 5-1 lead. Gauff had six double faults in the first set, including the final two points.
"It's never easy to start against such an aggressive player with a big serve," Badosa said. "I think I adapted pretty well in the [end] of the first set, and I played a pretty perfect tiebreak, so I'm really happy about that."
Badosa will play No. 7 seed Daria Kasatkina on Saturday.
Kasatkina, a 2021 finalist, recovered to defeat Aryna Sabalenka 4-6, 7-5, 6-0 in 2 hours, 23 minutes. Sabalenka, who was affected by an injury late in the match, had 22 double faults.
Kudermetova took the opening three games of the second set and led 5-1 before holding off No. 3 seed Ons Jabeur 7-6(5), 6-2. Kudermetova reached the semifinals for singles and doubles.
Rogers topped fellow American Amanda Anisimova 6-4, 6-4, improving her head-to-head record to 3-0. Rogers has not lost a set in three matches this week, while only dropping 18 games.