Djokovic, bidding for a seventh consecutive semi-final spot at Roland Garros, squandered two set points in the opener before losing the tiebreak and his game then fell apart.
Nadal, who turned 31 on Saturday and celebrated with two cakes, will next face another Spaniard, Pablo Carreno Busta, for a place in the semi-finals.
The 14-time major victor has dropped just 20 games in four matches as he strives to become the first man to win 10 titles at a single Grand Slam.
The Swedish legend has clear ideas about the Spanish derby between Pablo Carreno Busta and Rafael Nadal.
"In Rome it was not a good day for me".
Torrential rain forced Rafael Nadal and Novak Djokovic's quarter-finals at the French Open to be delayed until Wednesday.
Djokovic also became an 11-time quarter-finalist as he sauntered past Spain's Albert Ramos-Vinolas 7-6 (7/5), 6-1, 6-3.
"I have never had any issues with him before", Murray said. "It was just today it was just unusual how it came about".
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Meanwhile, Bautista Agut was trying to reach the last eight at a Grand Slam for the first time.
Heading into this French Open, many figured the women's draw was wide open, because of the absences of Williams' sister, Serena, and Maria Sharapova.
Second seed Karolina Pliskova advanced to round four with a 7-5, 6-1 victory over Germany's Carina Witthoeft.
Djokovic, who had needed five sets to beat Diego Schwartzman in the third round, was 3-0 down in the second set with Thiem's unerring accuracy finding the angles.
"It's a dream to beat Novak for the first time and reach the semi-finals at Roland Garros again", said Thiem, who was beaten by Djokovic in straight sets in the semi-finals in Paris in 2016.
In the next round, she will face unseeded Latvian Jelena Ostapenko. Third-seeded Simona Halep is playing No. 5 Elina Svitolina on Court Suzanne Lenglen.
That retirement sent Nadal into his record 10th Roland Garros semifinal and 25th Grand Slam semifinal, overall.
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