Fognini also defeated Mayer on Sunday in the Royal Guard Open final in Vina del Mar, Chile.
Almagro has won 12 ATP titles - all on clay - and six have come on the annual Latin American swing.
Top-seeded David Ferrer plays his second-round match on Thursday.
Fifth-seeded Marcel Granollers was upset by fellow Spaniard Pablo Andujar 6-3, 6-0. Andujar reached the quarterfinals.
Eighth-seeded Jeremy Chardy of France advanced to the second round, defeating Alejandro Gonzalez of Colombia 6-4, 7-5.
Other unseeded players advancing to the second round included Alexandr Dolgopolov of Ukraine and Daniel Gimeno-Traver of Spain.