Shelton, who is looking to become the first American man to win the title since 2003, has endured a disappointing run in this year's Grand Slams but arrived in New York with momentum after successfully defending his Canadian Open title earlier this month.
It was far from straightforward for the 23-year-old, however, as Griekspoor looked to follow up his stunning upset of top seed Alexander Zverev in Montreal.
"I started off a little bit nervous tonight, but I found my rhythm at some point through the second set," said Shelton, who is still hunting a first Grand Slam title.
"Obviously, a lot of credit to Tallon, he's a good player to play, he doesn’t give you a lot of rhythm. He has an attacking game style and I found things really difficult for the first set tonight."
The top American in the men's draw endured a woeful start under the Arthur Ashe Stadium lights, committing 14 unforced errors, including two double faults, and holding serve just once as Griekspoor raced through the opening set in less than half an hour.
The script flipped completely in the second, however, as Shelton found his range, sharpened his serve and broke three times to level the match.
The third set went to a tiebreak, with Griekspoor's errors proving costly, Shelton sealing the set when the Dutchman's sixth double fault of the match handed him the advantage.
Shelton then took complete control, storming to a 4-0 lead in the fourth before closing out the win to set up a second-round clash against Poland's Hubert Hurkacz.
US Open 2026
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