Seeking his 18th grand slam trophy, Nadal was stunned in a first set won to love by Thiem before the Spaniard came back to take the next two.
When asked if he suffered during the match, Nadal said: "That's the right word".
Nadal faces Juan Martin del Potro for a place in the final.
"First of all, I served really, really well today", Thiem said. "It was a great battle".
"Dominic is a fantastic player".
That mistake might have stayed in Nadal's head, because he played terribly in the ensuing tiebreaker.
Basilashvili claimed pre-match that he was playing his best tennis and he showed flashes early on, keeping his esteemed opponent at bay in one thrilling rally that brought him to the net.
His appearance in the quarter-finals at the US Open means he has reached at least that stage in all four Grand Slam tournaments this year, for the first time since 2011.
His 56 winners were erased by 59 unforced errors which in turn were three times as many committed by Nadal.
Though Thiem denied him at first, breaking back to get proceedings back on serve, an error-strewn game from the Austrian No. 1 allowed Nadal to level. Thiem's progress to the last-eight comes despite choosing to play European clay court events after Wimbledon before then losing his only summer hard court outing in Cincinnati to Stefanos Tsitsipas.
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The Spaniard saved a break point in games nine and 11 but he stood no chance in the tie break, losing it 7-4 following a routine forehand that he sent into the net, handing the set to his rival and setting up a decider after more than three and a half hours of awesome battle!
"On clay, it's one of the biggest challenges in sports to beat this guy".
The three-time US Open champion cruised into a two-set lead in little over an hour - ominous signs for Basilashvili, who had won one game against Nadal in their only previous meeting at the 2017 French Open.
In addition the Austrian found the slower conditions of the new Louis Armstrong Stadium much to his liking, reminiscent he said afterwards of clay, which is his favourite surface.
Basilashvili has had a strong showing in the tournament. It has been another magnificent season for Nadal, highlighted by an 11th French Open and Masters titles in Monte Carlo, Rome and Toronto.
The world number one came through a nerve-shredding sudden death tiebreak 7-5 to recover from a disastrous start and win 0-6 6-4 7-5 6-7 7-6 after four hours and 49 minutes of a baseline war.
Williams double-faulted on break point to send Kanepi up 5-2.
"The court, I think, made a big difference". "It gave me confidence".
He said: "Two matches in a row very tough ones". "They have to go back to Argentina and work again after this tournament. I needed to use all of it today".