In the other semifinal, 2016 runner-up Murray will face 2015 champion Stan Wawrinka in a matchup of three-time major title winners. Djokovic had won all five previous matches - and 11 of 12 sets - against Thiem, including in the French Open semifinals a year ago.
But Agassi left midway through the tournament due to what Djokovic had described as prior personal arrangements and the 30-year old from Belgrade was blown away by Thiem after beating Spaniard Albert Ramos Vinolas in his mentor's absence.
Djokovic won just eight points in the third set, which lasted a mere 20 minutes. Otherwise, he lost his No. 1 ranking to Andy Murray and lost in the third round of Wimbledon, the first round of the Rio de Janeiro Olympics, and the second round of the Australian Open.
While Thiem's part in Djokovic's downfall should not be overlooked, it was the Serbian world number two's astonishing collapse that was the talk of Roland Garros.
On a sunny Court Philippe Chatrier, she lobbed Garcia from the back of the court to give herself two match points.
He wasted two set points in the opener.
Former player Imaz, described as a "spiritual guru" was the squad's only survivor to welcome the arrival of Andre Agassi, who teamed up with Djokovic shortly before the French Open. Novak Djokovic effectively waved the white flag, bowing out of the French Open with a whimper on Wednesday.
Thiem is the only player to have beaten Nadal on clay this season, but the 23-year-old is under no illusion about the size of the task that awaits him in Friday's semi.
In Rome last month he beat Nadal in the quarter-finals only to be thrashed by Djokovic 6-1, 6-0. It's only the second time Djokovic lost a final set by the score of 6-0 in his 937 career tour-level matches.
Novak Djokovic's French Open title defense is in real trouble.
Known for tremendous footwork and court coverage, the No. 2-seeded Serb even stumbled and tumbled to the court, his racket flying out of his hands, early in the second set.
Since completing his career Grand Slam at the French Open 12 months ago, Djokovic has participated in four majors in a row without earning a trophy.
Both Nadal and Thiem have yet to relinquish a set.
Nadal led 6-2, 2-0 when Carreno Busta stopped. It was not immediately clear what was wrong with Carreno Busta.
The 20th-seeded Carreno Busta had taken a medical timeout after the opening set but continued to struggle with his serve, which he failed to hold all match. "To beat him for the first time in the quarters of the French Open is a dream".
Tuesday's rain meant all four men's quarterfinals were scheduled for Wednesday, with world No. 1 Murray facing Kei Nishikori and 2015 victor Stan Wawrinka confronting 2014 US Open champion Marin Cilic.
Simona Halep and Karolina Pliskova reached the women's semifinals.