Defending champion Nadal had suggested he would need to improve on his "very tough" third-round match with Karen Khachanov, where four sets and two tiebreaks were required, but there was little evidence that the seemingly tired world number one had lifted his game at Arthur Ashe Stadium.
Nadal has battled through his last two matches to reach the fourth-round in NY.
BBC Sport writes, "Basilashvili's approach was all or nothing.although unforced errors outstripped winners in the opening two sets before he began to find his range in the third".
Thiem is the first Austrian in the last eight at the US Open since Thomas Muster in 1996.
Isner came through a five-setter with Milos Raonic, who was hampered by a "sharp feeling" in his hip during the closing stages of the match.
The roof closed on the newly built Louis Armstrong Stadium for the first time during the second set, but it did not appear to disrupt the momentum of either player.
Isner, the 11th seed, is attempting to become the first home victor since Andy Roddick in 2003. "I'm able to compete with anyone out there, as I showed today". "This atmosphere is like a jungle".
Top-seeded Nadal followed a thrilling upset bid from Karen Khachanov in the third round with a 6-3, 6-3, 6-7 (8-6), 6-4 win over Nikoloz Basilashvili in the fourth round on Sunday. She was home, expecting the couple's first child.
The top-ranked Spaniard captured his 11th title in Paris by beating Thiem in straight sets in June. "Happy for him that he's in quarter-finals here", he said.
Nadal said: "Two matches in a row very tough ones".
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A matchup with Thiem would be a rematch of the French Open final, won by Nadal, and their first not on clay.
However, Basilashvili could not maintain this level in the fourth set as Nadal broke for a 4-3 lead which he would not relinquish.
Anderson was staring elimination from the event and needed a swift improvement in his play to keep his tournament hopes alive in the third set.
"But I'm really looking forward to it". At No. 32, Anderson was the lowest-ranked U.S. Open men's finalist since the ATP computer rankings began in 1973.
"Going to have my work cut out for me", said Anderson. "It wasn't meant to be". Thiem's only victory had come on clay, his best surface.
On the flip side, Anderson went 3-for-7 (43 percent) in break points in the match. "It's a dream coming true". "So I didn't face one break point, and I didn't feel so much pressure on service games", said Thiem.
Rafael Nadal has reached the quarter-finals of all four slams for the first time since 2011.
The Austrian ninth seed was denied a place in the last eight 12 months ago by a remarkable comeback from Juan Martin del Potro but exorcised his demons in impressive fashion to win 7-5 6-2 7-6 (7/2).
Nadal, the 2010, 2013 and 2017 champion and chasing an 18th major, believes the change from clay to hard court will not alter the dynamic Tuesday when he plays his eighth quarter-final at the tournament.