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The fifth seed, beaten by Nadal in the last two French Open finals, battled past the Spaniard 7-6 (7/3), 7-6 (7/4), 4-6, 7-6 (8/6) to deny him a crack at a record-equalling 20th Grand Slam title.
He will now meet German seventh seed Zverev who shattered the dreams of veteran Stan Wawrinka 1-6, 6-3, 6-4, 6-2 to book a maiden place in a Grand Slam last four.
Awaiting the winner of that clash will be either seven-time champion Novak Djokovic or six-time winner Roger Federer, who meet in the other semi-final.
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“Today I had the feeling I was lucky in the right situation. It is necessary because he is one of the greatest of all time. You need some luck to beat him.” Top seed Nadal had a 9-4 record over Thiem and had beaten him in all their five previous Slam meetings.
But the last time they played on hardcourts at the 2018 US Open quarter-finals it was a five-set marathon.
And the signs pointed that way again, with the opening set on serve to 2-2 before Thiem, gunning for a first Grand Slam title, worked a breakpoint but couldn’t convert.
Both baseline-huggers, it became a slugfest before Nadal managed to open some doors on the Thiem serve in game eight and he broke with a perfect lob from the back of the court.
But Thiem came roaring back, breaking back with a ripping crosscourt return. He saved a set point to take it to a tie-breaker where he stunned Nadal to seal a one-set lead.