The Italian soaked up everything Medvedev threw at him and emerged with a 6-3 6-7(4) 6-1 victory.
Sinner, the only player to win all three group matches, will face either world number one Novak Djokovic or number two Carlos Alcaraz in Sunday's final.
"It's an incredible feeling," Sinner said. "It was a really tough match today. I felt that he was playing more aggressively, especially in the first set. Somehow, I made the break and from that point I felt better."
If Sinner wins the title, he will receive $4.8 million as an undefeated champion, the biggest pay day in tennis history.
Medvedev was seeking to reach his 10th final of an impressive season and had the first opportunity to secure a break of serve at 1-1 in the opening set, but Sinner stood firm.
Sinner secured a decisive break out of nowhere in the following game as Medvedev let slip a 40-0 lead with errors flying off his racket.
Both players were rock-solid on serve in the second set with only one break point, saved by Sinner, on the way to a tiebreak.
Medvedev, sticking relentlessly to his heavy baseline game, grabbed the initiative and levelled the contest when Sinner sent a forehand over the baseline.
Sinner got his nose in front early in the decider when Medvedev double-faulted and Medvedev lost his focus as he became involved with a member of the crowd.
Sinner moved to the brink of victory with another break to storm into a 5-1 lead and the world number four brought up three match points with a superb backhand down the line and sealed victory at the first attempt.
After losing his first six matches against Medvedev, Sinner has now won three in a row against the Russian.