Nadal had a horror start to the match, with Thiem racing through the first set 6-0 in 23 minutes, just the second time in his US Open career that the Spaniard has lost a set 6-0.
Nadal recovered from losing the first set without winning a game to win 0-6 6-4 7-5 6-7 (4-7) 7-6 (7-5) in a gruelling four hour and-49-minute epic.
Remarkably, Thiem was the first top-20 player Nadal has faced at Flushing Meadows since his final win over Novak Djokovic in 2013, with last year's title seeing him take on a succession of unexpected opponents.
"I'm very, very sorry", Nadal revealed he'd said to Thiem as they embraced.
In an epic encounter that saw both players come up with some brilliant tennis, Nadal called on all his experience as he battled hot and humid conditions to triumph in a deciding set tiebreak after Thiem sent a smash long.
Thiem was 4-2 ahead in the fourth set before he was again reeled in and he was two points from defeat in the 12th game.
Rafael Nadal, trapped in a nightmare first set against Dominic Thiem in the quarterfinals of the US Open, knew where to look for answers. "I mean, he's a great guy".
"I'm so happy to reach another semi-finals in my favourite tournament", del Potro said in his on-court interview. That mistake might have stayed in Nadal's head, because he played terribly in the ensuing tiebreaker. "He's a fighter, he has a great attitude".
"To reach the semifinal again here in NY in my favourite tournament is very special to me", said Del Potro after his eighth win over Isner in 12 meetings.
Sevastova came into the match having broken her opponent's serve a tournament-leading 23 times.
"Sad for him because when arrive this moment, he did all the things well to win the match", Nadal said of Thiem.
"It's going to be stuck in my mind forever..." "Forever I'm going to remember this match". Yeah, it was [a] question of little bit of luck at the end. I think we nearly all the time have great matches.
That said, you don't win 17 Grand Slams without being able to dig deep in the face of adversity and after hanging in there, the top seed took advantage of a poor service game from the 25-year-old to earn the chance to serve out the second.
But while Isner was playing before what could count as a home crowd, del Potro got all manner of support throughout, from the blue-and-white flags or soccer jerseys dotting the stands to the repeated singsong chants of his nickname, "Delpo", punctuated by clapping.
The third set was another grind. Law and Whitaker say it feels like there's an irresistible force behind del Potro that's pushing him to the title.