Montanes, seeded fourth, prevailed over fellow Spaniard Ruben Ramirez Hidalgo 6-1, 7-6 (7-0) on the clay courts of the Buenos Aires Lawn Tennis Club.
Former Copa Claro champion Robredo moved on with a 7-5, 6-2 victory over Argentina's Maximo Gonzalez.
Also Monday, No. 8 seed Juan Ignacio Chela rallied to overcome Rui Machado of Portugal 4-6, 6-2, 6-2.
Italian Fabio Fognini and Argentine favorite Leonardo Mayer each notched first-day wins in Buenos Aires as well.