The world No. 2 had beaten the sixth seed in all five of their previous meetings, including a 6-1 6-0 win at the Italian Open in the build-up to Roland Garros, but was battered by Thiem on Suzanne-Lenglen.
The 31-year-old Nadal has spent only around eight hours on court, dropping 22 games in the process, to reach his record-extending 10th semi-final at the French Open.
The Spaniard, who is bidding for his 10th Roland Garros title, broke his opponent four times in the opening set and was leading 6-2, 2-0 when Carreno Busta was forced to quit just after 51 minutes in the quarter-final clash of the men's singles event.
A year ago, Djokovic became the first man in almost a half-century to claim a fourth consecutive major championship and completed a career Grand Slam at Roland Garros.
Even at this point you felt Djokovic could put up a fight and claw his way back into the match, but it wasn't to be.
The second-seeded Czech fended off three break points on her way to claiming the second set 6-3, clinching it with a forehand victor on her second set point.
"Well, obviously the way I turned the match around, it was not a good first set", Murray said. "I was also struggling the beginning of the second as well, managed to make some adjustments, changed a few things and got myself back into the match. It's unbelievable how hard it is to go deep in a grand slam because you have to play the best guys round after round and it's not getting easier on Friday".
Halep, who was runner-up at Roland Garros in 2014, will next face second-seeded Karolina Pliskova.
Against Thiem yesterday, Djokovic was out of sorts in many ways. "Just a pretty bad set", he told reporters.
Djokovic's backhand really let him down in the tiebreaker: All seven points won by Thiem ended with that stroke. He scored 38 winners as compared to Djokovic's just 18.
"More or less, all the parts of my game are kind of going up and down. That's what happened today".
Still, how unlikely was this result?
Plus, Djokovic had appeared in a record six consecutive semifinals in Paris. He split from coaches Boris Becker and Marian Vajda and other members of his team, bringing aboard Andre Agassi for Week 1 of the French Open. "At least I'm trying", Djokovic said.
"I know that I have achieved the biggest heights in this sport, and that memory and that experience gives me enough reason to believe that I can do it again".