Sao Paolo's Bellucci needed to win tiebreakers in the first and third sets to emerge with a hard-fought 7-6 (7-4), 5-7, 7-6 (7-1) victory over qualifier and fellow Brazilian Guilherme Clezar.
Bellucci was down 5-4 in the third set and but twice held serve, forcing a tiebreak and then dominating from there to shake off the 20-year-old challenger.
Italy's Fognini, meanwhile, didn't fare as well, falling to Argentine Guido Pella 7-6 (7-6), 1-6, 6-1.
Filippo Volandri of Italy -- last year's runner-up -- also moved on, outslugging Spain's Daniel Gimeno-Traver 6-3, 3-6, 7-6 (8-6) in first-round play.
2014: The Year in Music [PHOTOS]