World number one and defending champion Rafael Nadal meets Belgian David Goffin in the other semi-final later on Saturday to determine who will face Tsitsipas in Sunday's final.
Tsitsipas, 19, had pulled off the biggest win of his career on Friday before by stunning last year's finalist and world number seven Dominic Thiem in straight sets and was the underdog against Carreno Busta, who is ranked 52 places above him.
The youngster got off to a flying start as he broke his opponent in the second game and marched into a 4-1 lead. The Spaniard hit back with a break to level at 5-5 but wasted a chance to break again and lost his serve to concede the set.
Tsitsipas, ranked 63rd, broke the 27-year-old for a third time to pull 4-2 ahead in the second and eventually served out the match, collapsing to the ground in elation as he won on his second match point when Carreno Busta hit beyond the baseline.
Tsitsipas becomes the first Greek player to reach an ATP final since 1973 when Nicholas Kalogeropoulos made the final of the now defunct Des Moines Open.