Almagro, who made it to the fourth round of the Australian Open before losing in four close sets to Tomas Berdych, defeated Victor Hanescu 6-3, 6-7 (4-7) 6-3 to earn his third quarterfinal berth in four events this year.
Fernando Verdasco, the No. 3 seed, advanced by downing Javier Marti 6-4, 6-7 (2-7), 7-5 and that set up a quarterfinal duel with Albert Ramos.
The eighth-seeded Ramos survived against qualifier Igor Andreev 7-6 (7-4), 7-6 (7-4).
Beyonce flaunts bikini body, Blue Ivy in vacation pics
Jordana Brewster on Paul Walker: 'He was an enormous presence in my life'