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.
Notable deaths of 2014 [PHOTOS]