The top-ranked Djokovic cruised to a 6-2, 6-3, 6-2 victory over Griekspoor inside Arthur Ashe Stadium to move one step closer toward completing the first calendar-year Grand Slam in men's tennis since 1969.
"All in all very good, very good," said Djokovic, who is now 23-0 in Grand Slam tournaments this year. "I'm very pleased with the level of my tennis. All is going in the right direction."
The 34-year-old Djokovic will face 2014 U.S. Open finalist Kei Nishikori in the next round. The Serbian has won their past 16 matchups.
Also Thursday, No. 4 seed Alexander Zverev, No. 6 Matteo Berrettini, No. 7 Denis Shapovalov and No. 13 Jannik Sinner advanced to the third round.
10th-seeded Hubert Hurkacz, however, suffered an upset loss to unseeded Andreas Seppi, who won in four sets.
On the women's side at the U.S. Open, No. 1 seed Ash Barty earned a 6-1, 7-5 win over Denmark's Clara Tauson. Barty delivered 11 aces and accumulated 19 break points -- converting five -- en route to the dominant victory.
No. 16 seed Angelique Kerber, meanwhile, took little time to defeat unseeded Anhelina Kalinina 6-3, 6-2 to reach the third round. Kerber will face 2017 champion Sloane Stephens in the next round.
The only seeded woman to lose in the second round was No. 24 Paula Badosa, who fell to Varvara Gracheva 6-4, 6-4.
Fourth-seeded Karolina Pliskova, No. 6 Bianca Andreescu, No. 7 Iga Swiatek, No. 10 Petra Kvitova, No. 11 Belinda Bencic and No. 14 Anastasia Pavlyuchenkova each advanced.