The world number one had hoped to equal Roger Federer's record of 20 grand slam singles titles but that will have to wait after fifth seed Thiem powered his way to a 7-6 (3) 7-6 (4) 4-6 7-6 (6) victory to set up a clash with Alexander Zverev.
Zverev had joked of his rivals: "I'll have a cold glass of Coke sitting in my hotel room with my AC (air conditioning) hopefully watching them play for six hours."
It did not quite take that long but after four hours and 10 minutes it was Thiem who clinched another dramatic Melbourne encounter.
He looked like he might have blown his chance when he played a horribly nervous game serving for the match at 5-4 in the fourth set but recovered impressively in the tie-break.
Nadal saved two match points, the second with a HawkEye challenge after a Thiem lob had been called in, but on the third the Spaniard netted a forehand.
Thiem, beaten by Nadal in the last two French Open finals, had never reached the last eight here before but, if he can maintain this form, he will have a good chance of winning a first grand slam title.
Nadal would have expected a fierce battle, having taken nearly five hours to see off Thiem in their only previous grand slam meeting on hard courts at the US Open in 2018.
The 26-year-old has improved significantly since then and he stunned Nadal time and again with his power off forehand and backhand, his court coverage and the way he was able to recover from seemingly desperate positions.
Thiem has surely never played better yet Nadal will be hugely disappointed that he did not capitalise on leads he had in both the first two sets.
The Spaniard had wobbled in sight of the finish line against Nick Kyrgios on Monday, but his opponent had not been able to make him pay.
Thiem was much more ruthless. After breaking for 5-3, Nadal attempted to serve out the opening set and had one set point but could not take it and back came the Austrian.
One of Nadal's greatest strengths has been his mental fortitude at the biggest moments, yet there were more errors in the tie-break.
The top seed played more aggressively at the start of the second set and got his reward with a break for 3-2, but a time violation in the next game left him furious, Nadal saying to umpire Aurelie Tourte: "You don't like the good tennis."
He lost his rhythm completely, becoming increasingly agitated with a malfunctioning fan next to his seat and allowing Thiem to break back for 4-4 with a meek double fault.
Nadal saved one set point at 5-6 and fought back from 0-4 to 4-4 in the tie-break only to play a dreadful drop shot two points later, giving Thiem the impetus to take the set, with a little help from the net cord.
Competitors do not come more determined than Nadal but it is 13 years since he won a match from two sets down.
He took a step towards it by breaking Thiem to win the third, crouching down and fist-pumping wildly in celebration.
Nadal had three chances to break again at the start of the fourth but Thiem dug in just when it looked like the momentum had really turned and wrestled it back in his direction.
Earlier, Zverev reached his first grand slam semi-final with a 1-6 6-3 6-4 6-2 victory over Stan Wawrinka.
So much has been expected of the German for so long but it was not until he came to a slam a little under the radar that he has been able to make his big breakthrough.
Zverev had a difficult 2019 and arrived in Melbourne with his serve apparently in crisis and having lost all three of his matches playing for Germany at the ATP Cup.
The 22-year-old, who has promised to give all his prize money to wildfire relief efforts should he win the tournament, said: "I was very impatient. I was paying too much attention to them. The grand slams maybe meant too much for me.
"This year I actually came into the Australian Open with absolutely no expectations because I was playing horrible."
Wawrinka, the champion in 2014, rated his gruelling fourth-round victory over Daniil Medvedev the best he had played since knee surgery two and a half years ago but this felt like a match too far after an explosive start.
The 34-year-old said: "The fact that I wasn't moving my best after a set didn't help me, especially against a player so solid like him. In general I'm happy with the tournament. It's positive for the rest of the year."