The 2009 and 2011 winner is not seeded for the ATP 250 event this year but only needed 74 minutes to end the challenge of the Austrian qualifier on Tuesday.
Both players performed well with breaks of serve a rarity, but when they came Bellucci converted two of his three while saving the three break points Melzer created.
The Brazilian finished the opening set with three consecutive games and repeated the same feat in the middle of the second to earn his victory.
He will be joined in the second round by Robin Haase after the seventh seed survived a comeback from Aleksandr Nedovyesov to progress 6-4 6-7 (3-7) 6-3.
The Dutchman will have to wait to find out his second round opponent after Henri Laaksonen and Gianni Mina saw their match suspended with the former leading 7-6 (7-5) 4-4 when darkness stopped play.
Yann Marti will meet second seed Marcel Granollers in the next round after his hard-fought 5-7 7-6 (9-7) 7-6 (7-5) win over Daniel Gimeno-Traver.
Resuming at 5-5 in the final set, Marti – who saved three match points in the second set – battled to victory.
Top seed and defending champion Mikhail Youzhny awaits Kenny De Schepper after the world number 110 beat Filippo Volandri 6-3 7-6 (7-3).
Blaz Rola reached the second round of an ATP-level clay-court tournament for the first time as he saw off the challenge of Fabiano De Paula 6-4 6-4.
The Slovenian matched his performance at Wimbledon with a dominant service performance, while also winning 50 per cent of the break points he created on the Brazilian's serve.
Meanwhile, Pablo Andujar and Jan-Lennard Struff were both gifted a place in the second round after their respective opponents Gilles Simon and Andreas Haider-Maurer retired in the second sets of their matches.