The world number one looked like he would cruise to an easy win after pocketing the first two sets but the Georgian stepped up his game to take the third.
The U.S. Open champion would not be denied, however, and fired his seventh ace on match point to end the three hour, 20-minute tussle.
"Two matches in the row that were very tough ones," said Nadal, who had to battle back against Russian Karen Khachanov two days ago.
Nadal praised the 26-year-old Basilashvili, who will break into the top 30 for the first time when the new rankings are released after the tournament.
"He played fantastic. A lot of credit to him. He was playing great, hitting the ball very, very strong," the Spaniard said.
"I'm very happy to be through," added Nadal after setting up a rematch of June's French Open final, when he crushed Thiem in straight sets to win his 17th major.
Basilashvili only managed to convert two of his seven break point opportunities and committed 59 unforced errors to Nadal's 19 on a partly cloudy day at Arthur Ashe Stadium.
Despite Sunday's loss the result is big step forward for Basilashvili, who suffered a 6-0 6-1 6-0 drubbing by the claycourt king when they met at the French Open in June.
Nadal said he will need to raise his game to get past the red-hot Thiem, who reached his first U.S. Open quarter-final when he defeated 2017 U.S. Open runner-up Kevin Anderson in straight sets.
"He's a fantastic player. He's a very powerful player. He's a great guy," Nadal told reporters.
"I have a very good relationship with him. Happy for him that he's in quarter-finals here," he said.
"I'll need to play my best match of the tournament."
(Reporting by Rory Carroll, editing by Pritha Sarkar and Ed Osmond)
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