The Italian won 6-0, 5-7, 7-5, and in the final will face Novak Djokovic, who ended his run of semi-final losses as he cruised past Yannick Hanfmann 6-3, 6-4.
Musetti is ninth in the race to qualify for the eight-man Finals which start on Monday in his home country. He needs to win the Athens tournament to collect enough points to overtake Canadian Felix Auger-Aliassime in the standings.
Musetti started well, dropping just five points as he swept to the first set.
In the second set, Musetti broke in the seventh game but Korda broke straight back and then broke to love in the decisive 12th game.
Korda had the first break point, and a match point, in the 10th game of the third set.
Musetti escaped, ending a long rally by drawing the American to the net with a drop shot and then lobbing him.
The Italian immediately broke, then saved a break point in his next service game before breaking Korda again to win in two hours 20 minutes.
“I think we both played a fantastic match,” Musetti said. “The level was getting higher and the adrenaline was intense.”
“Achieving the final here is really important and I am already focusing on tomorrow.”
Earlier, Djokovic took 79 minutes to dispatch German qualifier Hanfmann.
“I think it was the best tennis I’ve played this tournament,” Djokovic added. “It came at the right time. Hanfmann poses a great threat because he serves big, has a big game, so I needed to really stay focused.”
Djokovic broke his four-match semi-final losing streak. He had fallen in the last four at Roland Garros, Wimbledon, the US Open and Shanghai.
His last semi-final victory was at the Geneva clay-court event in May when he went on to beat Hubert Hurkacz in the final for his 100th career singles title. The only men with more are Roger Federer (103 titles) and Jimmy Connors (109).
Djokovic was roared on by a noisy, supportive crowd at the event which this year took the place of the short-lived Belgrade Open on the ATP schedule.
“Thank you for filling out this amazing stadium again,” Djokovic told the fans. “I’ve played in some of the most beautiful indoor arenas in the world, but I can definitely say this is one of the top three I’ve ever played in.”