As clouds moved into the Gulf, conditions remained hot at 33 degrees Celsius at the Aviation Club, where Berdych tamed Italian qualifier Thomas Fabbiano 6-2, 6-2.
Bautista Agut claimed a 16th win of the season with a 6-0, 6-1 hammering of South Korean youngster Chung Hyeon.
Flamboyant Australian Nick Kyrgios won his second match in less than 24 hours, after a late arrival in the emirate following his first career title last weekend in Marseille.
The unseeded threat dispatched Kazakh Mikhail Kukushkin 6-3, 6-2 and next faces Berdych, who he beat last weekend in the French indoor semi-finals, although two-time finalist Berdych knocked the talented 20-year-old out of the Australian Open last month.
“He’s doing well, for me it’s going to be a big challenge. I have a lot to get back after last week. I must refocus myself, try to raise the game, and play some good tennis again,” Berdych said.
“He’s full of fireworks on the court – in a good way. He’s playing really aggressive. He’s going for a lot of shots, risking a lot.”
Bautista Agut, who has won two titles already this season, will next face Marcos Baghdatis after the popular Cypriot defeated Canadian Vasek Pospisil 6-4, 6-2.
The 18th-ranked Spaniard is thrilled with his progress this season after schooling the teenaged Chung, denying him a second career ATP quarter-final.
“It’s good to win like this, playing well and doing everything you practise for. But I have to be humble and know that to keep winning like this, it’s impossible.
“For every win you must work a lot, play for two hours. I’m happy because I won fast today (54 minutes) but I know it’s going to be tougher and tougher.”
Two more seeds also reached the last eight, with sixth seeded Spaniard Feliciano Lopez beating Czech Lukas Rosol 6-3, 7-6 (9/7) and German eighth seed Philipp Kohlschreiber putting out young Croatian Borna Coric 6-2, 7-6 (7/5).
