Almagro broke open a tight first set and then ran away by taking the first five games of the second in a 6-4, 6-1 win over Tobias Kamke.
Almagro lost his serve one in each set but he was up 5-0 in the second set by the time the second break occurred. He won 56 percent of the points on serve but took 63 percent when Kamke served.
Wild-card entrant Julian Reister ousted No. 5-seeded Fernando Verdasco 6-2, 6-3 in just more than an hour. Reister lost 11 points in his nine games serving. Lukas Rosol advanced by eliminating eighth-seeded Victor Troicki 5-7, 7-3, 6-4.
Sixth-seeded Philipp Kohlschreiber didn't lose his serve after the first set of a 4-6, 6-2, 6-3 decision over Bjorn Phau and Florian Mayer, the seventh seed, edged past Horacio Zeballos 7-6 (9-7), 7-5.
Also Tuesday, Fabio Fognini defeated Marsel Ilhan 7-5, 6-2; Jeremy Chardy beat Albert Montanes 6-4, 6-2; Albert Ramos romped over Pablo Andujar 6-1, 6-4; and Tommy Haas, another wild card, rolled past Martin Klizan 6-2, 6-1.