The 19-year-old won 78% of his first-service points and struck 24 winners in his 7-5, 6-0, 6-3 victory over Goffin to earn a place in the second round in Paris.
In the victory, which took just under two hours, Sinner won 11 straight games from 5-5 in the first set to a 3-0 lead in the third set.
The win was his second successive over Goffin, having beaten him at the ABN AMRO World Tennis Tournament in February.
But he showed no hint of any nerves on the big stage, breaking first to race into a 3-1 lead under the closed roof on the new Court Philippe-Chatrier.
“The first set was very tight, a lot of breaks of serve, and it was the key to manage to win the service games quite easily. In the beginning, that was not easy. He was returning well. I was not serving that well. But the balls here are very heavy. The court was heavy. It was not easy. But when you win the first set and go up a break, it’s a little bit easier to play. I mean, he’s a Top 15 player, so you always have to be focused. I tried to do that, and today it worked.”
“It’s tough to lose a match like that, but he’s a great player. When I saw that I was playing against him first round, I knew it was going to be tough, especially in the first round when you have no rhythm. I didn’t have a win on the clay.”
Sinner will next face French qualifier Benjamin Bonzi who defeated Finland’s Emil Ruusuvuori.