Murray got off to an underwhelming start in his first service game as the Brazilian put him under pressure, although he managed to fend off the 27-year-old's challenge.
However, Bellucci did secure the break in the Scot's second service game following an error from Murray, but the number three seed hit back immediately as he upped the ante with his groundstrokes across the court, while a double fault from the Brazilian handed him the break.
Murray found his best form as the set progressed and he secured another break to allow him to take the first set 6-3.
Bellucci struggled to find his rhythm in the first game of the second set and the British number one punished him by securing a break, capitalising on a double fault from the 30th seed.
The 28-year-old put further pressure on the Brazilian to yield another break and he managed to secure the second set 6-2.
In the third set, Murray carved out another opportunity to break when he created two break points, but Bellucci battled back strongly to stave off the Scot to secure the hold.
However, the number three seed picked his moment to turn up the aggression and secure the break in the third to take a 6-5 lead, before serving out the match to book his place in the last 16 of the competition.
Murray will face Kevin Anderson in the fourth round.