Monfils, seeded seventh, topped Robin Haase 3-6, 7-6 (7-3), 6-2. Monfils won one more point (93-92) over the course of the match but took 60 percent of the points in the third set, when he recorded three of his five breaks on the day.
Almagro, the No. 8 seed, needed nearly two hours in getting past Pablo Andujar 7-5, 7-6 (7-1). Almagro was good on just 46 percent of his first serves and survived despite dropping serve three times.
The rest of the matches Tuesday were first-round affairs and included wins by two seeded players and an upset. No. 9 Richard Gasquet defeated Juan Ignacio Chela 2-6, 7-6, (7-2), 6-4 and 15th-seeded Milos Raonic swept by Radek Stepanek 6-4, 6-2, but Nikolay Davydenko capped the day's action with a 6-4, 7-6 (8-6) decision over 10th-seeded Alexandr Dolgopolov.
Also advancing Tuesday to the second round were Daniel Gimeno-Traver, Ivan Dodig, Simon Greul, Feliciano Lopez, Kei Nishkori, Carlos Berlocq and Juan Carlos Ferrero.
Celebrity Families of 2014 [PHOTOS]