The 23-year-old British number one recovered from a slow start to win 3-6 6-1 6-4 6-2 under the roof on Court Suzanne-Lenglen.
Draper joins the rest of the Brit pack; Emma Raducanu, Katie Boulter, Sonay Kartal, Jacob Fearnley and Cameron Norrie, in the second round - the first time the nation has had six first-round winners at Roland Garros since 1973.
It is a far cry from last year, when Draper was among the six first-round losers, as Britain suffered a depressing total wipe-out.
Draper seemed to be caught cold at the start of the match, with world number 68 Bellucci audaciously wrapping up the first set with an underarm serve.
But Draper, up to four in the world in the live ATP rankings, eventually found his rhythm and eased to his maiden victory at the Paris grand slam in two hours and 35 minutes.
"Bellucci came out and played really aggressively from the start, playing some amazing shots and caught me a little bit off guard," he said.
"I'm proud of the way I managed to turn the match around.