Fourth-seeded Goffin's aggressive forehands helped him save two break points at 4-4 but he missed another one to hand the 20-year-old Tsitsipas a decisive break.
"I fought very hard and gave my soul out on the court, which is the biggest pleasure in this sport," top seed Tsitsipas said.
"I was saying to myself while returning that I needed to stay aggressive, I needed to press as much as I could because otherwise I had no other option.
"I had to turn the whole thing around... risking a little bit more and I think at the end it paid off."
Tsitsipas will face Uruguayan veteran Pablo Cuevas, who outlasted Spanish teenager Alejandro Davidovich Fokina 3-6 6-2 6-2, in Sunday's final.
(Reporting by Hardik Vyas in Bengaluru; Editing by Christian Radnedge)
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