Austria's Thiem knocked out South African fifth seed Kevin Anderson - the man who lost in the final to Nadal at Flushing Meadows previous year - in three sets earlier on Sunday.
"It's a big victory for me", Nadal, the champion in 2010, 2013 and 2017, said.
Shapovalov is 19, making him the youngest player in the ATP top 100. The most revealing statistic was Anderson's 14 unforced errors to the five of Thiem, which saw the world number nine break serve twice to take the set in a convincing 6-2 scoreline.
The Spanish star cruised into a two-set lead, seemingly fully recovered from the trauma of fighting back from a set and break down to Russian Federation s Karen Khachanov on Friday.
But Basilashvili stunned Nadal in the third set tie-break and even retrieved a break for 3-3 in the fourth set before Nadal's greater composure paid off.
Instead Nadal is facing Dominic Thiem who already has an upset under his belt.
Dominic Thiem has eliminated 2017 U.S. Open finalist Kevin Anderson with a 7-5, 6-2, 7-6 (2) victory in the fourth round.
Del Potro seized control by breaking to win the 10th game and never looked likely to relinquish his grip on the contest, particularly after Coric received treatment on an apparent hamstring problem late in the second set. He disposed of Basilashvili 6-3, 6-3, 6-7 (6), 6-4 on Sunday.
Pliskova, the three-time WTA tour ace leader who briefly ascended to world number one a year ago, has surprised Williams before at the US Open, beating her to reach her first Grand Slam final in 2016.
Thiem has unfinished business at the US Open. "It's a dream coming true".
The 32-year-old has certainly been pushed in his last two matches, both of which have gone to four sets.
The French Open has been by far Thiem's best grand slam and beating Anderson was a major breakthrough.
Today, they will try it for the first time in the US Open quarter-finals at the Billie Jean King National Tennis Centre and see how it goes.
Thiem had never passed the fourth round at Flushing Meadows, but the Austrian, who turns 25 on Monday, served almost perfectly and gave the 6ft8in South African little time to unleash his mammoth groundstrokes.
The story of the match was Thiem's ability to nullify Anderson's big serve.
But for all their history, Rafael Nadal and Dominic Thiem have never played on hard court, and they've never played in NY. "I hope that it's a little bit more comfortable on hard court". But Nadal is more vulnerable in NY than he is at Roland Garros: At the French Open, he has an 86-2 career record and 11 titles. "He hits the ball strong and has a big serve", Nadal added. He made life pretty hard for me.
"I thought it was great tennis throughout", Anderson said.
Giants collide when third-seeded del Potro - who claimed his lone Grand Slam title at the 2009 US Open - takes on Isner - the last American main remaining in the draw - for a semi-final berth. "Constant noise going on the whole time, which obviously as tennis players, it's nice to have quiet".