Haas, playing in South America for the first time since 2002, defeated Italian Potito Starace 7-6 (7-2), 6-3 on the clay courts at Sao Paulo.
The German veteran, who reached the finals in Zagreb this month, is coming off a second-round exit at Delray Beach last week.
His next opponent will be Horacio Zeballos, who dispatched fellow Argentine Guido Pella 6-3, 6-4.
The news wasn't as good for the second-seeded Almagro, who was upended by Argentine Federico Delbonis 3-6, 6-3, 6-2. The Spaniard won the Sao Paulo event in 2008, 2011 and 2012.
Delbonis goes on to face Spain's Albert Montanes, who was leading Joao Souza 6-4, 2-3 when the Brazilian retired with muscle problems.