World number one Nadal saw off a battling Nikoloz Basilashvili, the first Georgian in a fourth round in New York, 6-3, 6-3, 6-7 (6/8), 6-4.
“It’s a big victory for me,” said the 32-year-old Nadal, the champion in 2010, 2013 and 2017.
“He was hitting the ball very strong and when I thought I was in control of the points, he always came back with strong shots.”
It was world number 37 Basilashvili’s strength, however, which proved his undoing.
His 56 winners were erased by 59 unforced errors, three times as many as Nadal.
Elsewhere, Juan Martin del Potro, the 2009 champion, beat Borna Coric 6-4, 6-3, 6-1 to make the last-eight for a third successive year.
The third seeded Argentine will next face John Isner, the last American man standing, who defeated Milos Raonic 3-6, 6-3, 6-4, 3-6, 6-2, taking his tournament aces total past the 100-mark.
Thiem, who beat Nadal on clay in Madrid in the run-up to Paris, booked his place in the quarter-finals for the first time with a 7-5, 6-2, 7-6 (7/2) win over 2017 runner-up Kevin Anderson.
The ninth-seeded Austrian’s three other appearances in the last-eight of a Slam were all on Paris clay in 2016, 2017 and 2018.
Nadal leads their head-to-head 7-3 with all of their matches coming on clay courts, including in the final of Roland Garros in June when he captured an 11th French Open.
“Thiem is a great opponent, he has the big serve and he had a big win today against Kevin,” said Nadal who has made the last-eight of every Slam in the same year for the first time since 2011.
Last year, he had two match points against del Potro in the fourth round but fell in five sets.
‘Against Rafa? I’m not sure’
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.
He is the first Austrian in the last-eight at the US Open since Thomas Muster in 1996.
“Against Rafa, I’ve had three very nice experiences and seven horrible experiences,” said Thiem who turns 25 on Monday.
“On clay, it’s one of the biggest challenges in sports to beat this guy. I hope that it’s a little bit more comfortable on hard court, but I’m not sure.”
Third-seeded del Potro made the last-eight for the sixth time against Croatian 20th seed Coric who was playing in his maiden last 16 of a Slam.
Coric, 21, saw his challenge undone by a right thigh injury which needed two visits by the trainer as well as general unhappiness with the humidity of the Arther Ashe court.
“To win in straight sets in every round is good for the body and physical condition,” said del Potro.
“But every round is a battle. I just hope to keep my level high against John in the next match.”
Isner reached the quarter-finals for the first time in seven years and hailed the patriotic tournament atmosphere as a “jungle.”
“There were a lots of ups and downs in the match. The atmosphere was like a jungle,” said Isner of the crowd on the new Louis Armstrong Stadium enjoying a raucous Labor Day weekend.
Isner, 33, who will become a father for the first time in two weeks, said he will not fear del Potro even though he has lost to the Argentine star seven times in 11 meetings.
Their most recent clash saw del Potro triumph in straight sets in the fourth round at Roland Garros this year.
“Why not me? Let’s keep going,” said Isner after firing 56 winners past Raonic and 20 aces taking his ace count to a tournament-leading 112. —AFP