Eight third-round matches were scheduled for Thursday but only three were completed before rain led to a suspension of play. Organizers planned to finish the third round and complete the quarterfinals Friday.
Among those who had their matches delayed Thursday was second-seeded Rafael Nadal, who will now need to win twice Friday to have a chance at a third consecutive Barcelona Open title.
Almagro advanced with a 6-3, 6-4 victory over 15th-seeded Marcel Granollers. Almagro fought off six of the seven breakpoints he saw while getting three breaks against Granollers.
He'll play seventh-seeded Juan Monaco in the quarterfinals after Monaco beat 10th-seeded Jeremy Chardy 6-0, 7-6 (13-11). Monaco won nearly three-quarters of the points in the first set but needed to win a lengthy tiebreaker to get out of the third round.