Bautista Agut, in his first match against a Top 10 player, came away a 7-5, 2-6, 6-3 winner over the world No. 6 Berdych. Each player won 95 points on the day but Bautista Agut had a seventh-point edge in the third set.
Berdych had two double faults, with the second giving Bautista Agut a break and 5-3 lead in the third set.
Fifth-seeded Benoit Paire eliminated Marin Cilic, seeded third but 32 spots higher in the world rankings, 4-6, 6-1, 7-5. Cilic managed at least one break in each set but Paire also had four breaks, including two in the decisive third set.
Aljaz Bedene upset fourth-seeded Stanislas Wawrinka 6-2, 7-6 (8-6). Bedene dropped serve once while winning 70 percent of the points when serving.
More to form, second-seeded Janko Tipsarevic handled No. 8-seeded Go Soeda 6-2, 6-4 with five breaks.
Tipsarevic goes against Bedene in Saturday's semifinals. Tipsarevic won the only other time they met.
Saturday will be the first meeting between Bautista Agut and Paire.