The final that fans are hoping to see next weekend between Sinner and top seed Carlos Alcaraz seemed in danger when big-hitting Dutchman Griekspoor took the opening set and was serving for a 6-5 lead in the second.
But second seed Sinner, displaying some of the steel that has made him a Grand Slam champion, came through in the clutch by securing his first break of the match when he needed it most to go up 6-5 and then held serve to level the contest at 1-1.
After flirting with danger, the Italian took command by breaking the 26th-ranked Griekspoor at the first opportunity in the third set and broke a second time to go 5-1 ahead before serving out the match with a forehand winner into an open court.
"I think even the first set was really close, but it went away,” said Sinner, who will next play the winner of the match between Chris O’Connell and Martin Damm.
"Mentally I tried to stay strong, which I did, and I was in a tough situation today. It’s been a tough day for me, but very happy about the result."
There was no escape for sixth-seeded Dane Holger Rune who was swept aside 6-1, 6-1 by Fabian Marozsan in 59 minutes.
The Hungarian, known for his delicate touch and one of the best drop shots in the game, struck 23 winners and converted all five of his break opportunities while Rune was 0-3 on his chances.
Ranked 57th, danger man Marozsan is quickly developing a reputation as a giant killer, having last year knocked off then-world number two Alcaraz in the third round of the Rome Masters, which was widely rated as the upset of the year.
The 24-year-old also took down eighth seed Casper Ruud at the Shanghai Masters, reaching at least the last 16 in all three Masters 1000 events he has played. Blocking Marozsan's path to the fourth round is Australian Alexei Popyrin, who advanced with a 6-4 6-4 victory over Czech Jiri Lehecka.
"It is a great feeling to play at the highest level," Marozsan told reporters. "I try to play my best tennis and today against a top 10 player and I am enjoying my time here."
After Americans Taylor Fritz, Tommy Paul and Frances Tiafoe all exited in the second round, Ben Shelton is the top-seeded US player left in the draw, the 16th seed making the third round with a 6-3, 6-4 win over Spanish wildcard Martin Landaluce.
