Nadal led 6-2, 2-0 when Carreno Busta stopped.
Nadal rolled through the first set in 32 minutes, chiding himself for two service lapses while monstering Carreno Busta.
It was 12-times grand slam champion Djokovic's first defeat before the semi-finals in Paris since 2010 when he was also beaten by an Austrian, Juergen Melzer.
On resumption of play Wednesday morning following a rain delay, the world no 2 was unable to cope with the power of Thiem, who hit 38 winners in contrast to Djokovic's 18.
"The first set, I could easily lose, and I think it was the key for the first victory against him to win this close first set, and then have a good start to the second set, as well", said the Austrian.More news: Eurozone growth revised up to 2-year high
"This was obviously was not the ideal way and especially against a good friend".
Top-ranked Andy Murray is through to the French Open semifinals after beating eighth-seeded Kei Nishikori 2-6, 6-1, 7-6 (0), 6-1.
He had only won one set in all their five previous meetings and lost easily to Djokovic here in the last four 12 months ago. Romanian third seed Halep trailed by a set and 5-1 against the hard-hitting fifth seed but somehow dug herself out of a hole before storming to victory to set up a semi-final against Czech second seed Karolina Pliskova. Bopanna and Dabrowski survived a tough challenge from Hlavackova and Edouard in the opening set before dominating the second set to earn a shot at the title.
Pliskova reached the U.S. Open final a year ago, but the Czech player had never been past the second round at Roland Garros in five previous appearances.
Nadal makes a 10th RG semifinal.
Thiem was aggressive throughout, at times overtly so, but stuck to his guns and continued in the same vein in the next two sets. I only care that I'm in the semi-finals.More news: How Verizon hopes to grab digital ad dollars with Yahoo
Thiem, who plays with a single-handed backhand, said he needed to improve the defensive side of his game after struggling against the ferocious top-spin on Nadal's forehand during their last few encounters.
He said: "When have you seen Djokovic lose 0-6 in a Grand Slam quarter-final? I have the belief that I can reach them again", added Djokovic.
Svitolina, who had been two points from defeat in the fourth round against Petra Martic, finished the quarter-final with 45 unforced errors. "At least I'm trying", Djokovic said.
Thiem took the momentum of winning the opener into the second and moved 3-0 up in no time at all.
"He deserved to win, he was definitely the better player on the court".
"But all of the tournament I have played well and despite the weather conditions, I played well".More news: Ariana Grande Resumes Tour After One Last Tribute to Victims