Almagro, the 2011 winner, disposed of Argentina's Guido Pella 6-3, 6-2 in a quick match at Buenos Aires Lawn Tennis Club.
Brazil's Bellucci, however, didn't fare as well in another opening-day contest as little-known Argentine wild card Diego Sebastian Schwartzman bounced him 6-4, 4-6, 6-1.
Fifth-seeded Horacio Zeballos of Argentina also fell in a surprising result as Aljaz Bedene of Slovenia bested him 4-6, 7-6 (7-4), 6-3.
No. 8 seed Albert Ramos of Spain and Argentine wild card Federico Delbonis also recorded wins Monday.