The top-seeded Ferrer dispatched Colombia's Santiago Giraldo in straight sets, 6-4, 7-6 (7-1), at Buenos Aires Lawn Tennis Club.
Meanwhile, No. 4 seed Almagro moved on with a comeback 1-6, 7-5, 6-4 decision over fellow Spaniard Daniel Gimeno-Traver.
Ferrer, a wild card in the Argentina event, will next face Spain's Albert Ramos, who posted a 6-2, 6-1 triumph over Italy's Filippo Volandri.
Also advancing Thursday was No. 8 seed Jeremy Chardy of France, who battled to a 6-3, 3-6, 7-6 (7-3) victory over Ukrainian Alexandr Dolgopolov.