Delbonis, an Argentine, won 7-5, 7-6 (2) against Bellucci, a Brazilian, who also took the clay-court title at Swiss ski resort Gstaad in 2009.
Delbonis will continue rising in the rankings after victory over Federer in the Hamburg semi-finals helped him climb from No. 114 to 65.
Federer is top-seeded this week and has a second-round match scheduled on Thursday.
Four Spaniards had first-round wins on Tuesday.
Eighth-seeded Roberto Bautista eased past Henri Laaksonen of Switzerland, 6-2, 6-3; Daniel Gimeno-Traver beat Kenny De Schepper of France, 6-4, 6-4; Guillermo Garcia-Lopez beat Adrian Ungur of Romania, 6-4, 6-7 (2), 6-1; and Pablo Andujar eliminated Victor Crivoi of Romania, 6-1, 6-2.
Results from day one of the Swiss Open in Gstaad (x denotes seeding):
Robert Bautista Agut (ESP x8) bt Henri Laaksonen (SUI) 6-2, 6-3
Daniel Gimeno-Traver (ESP) bt Kenny De Schepper (FRA) 6-4, 6-3
Pablo Andujar (ESP) bt Victor Crivoi (ROM) 6-1, 6-2
Guillermo Garcia-Lopez (ESP) bt Adrian Ungur (ROM) 6-4, 6-7 (2/7), 6-1
Federico Delbonis (ARG) bt Thomaz Bellucci (BRA) 7-5, 7-6 (7/2)
Victor Hanescu (ROM) bt Dustin Brown (GER) 7-6 (8/6), 6-2
Robin Haase (NED) bt Joao Souza (BRA) 6-4, 7-6 (9/7)
Daniel Brands (GER) bt Marco Chiudinelli (SUI) 6-3, 3-6, 6-2