Thiem seized the early momentum inside Arthur Ashe Stadium, handily carrying the first set without dropping a single first-serve point, but a double fault and a forehand error saw him hand Medvedev the break and the early lead in the second.
The Australian Open runner-up spent the remainder of the frame playing catch up, converting on a critical break to level the score 5-5 before the 24-year-old Russian dragged it into a tiebreak.
The 27-year-old Austrian, who had a trainer examine his ankle before the third set, slipped and fell twice in the match, unfurling a string of frustrated exclamations in the second incident, but regained his cool to survive yet another tiebreak to clinch the affair with a power forehand and a roar.
“I played my best tennis then towards the end of the sets,” said Thiem, who is gunning for a maiden Grand Slam title. “Both tiebreaks were amazing. Tiebreaks are mentally a tough thing. I don’t like them at all to be honest. I’m really happy to be through. It was a great semi-final,” he added.
Thiem will take on Germany’s Alexander Zverev in Sunday’s U.S. Open final.