Almagro saved a match point in the second set before downing Minar 1-6, 7-6 (10-8), 6-4. It was Minar's first appearance of the year in an ATP event.
Daniel Gimeno-Traver set up a quarterfinal meeting with Almagro by defeating qualifier Alessandro Giannessi 6-4, 6-3.
Also advancing Thursday were No. 3 seed Albert Ramos, who defeated Evgeny Korolev 6-3, 6-3, and sixth-seeded Grigor Dimitrov, who was a 6-3, 6-4 winner over Frederico Gil.
Robredo is a two-time winner of this tournament and has recently returned from a lengthy absence due to injury.