A day after being taken to three sets by Czech qualifier Tomas Machac, Djokovic was back to his ruthless best on Wednesday, blasting past a shell-shocked Griekspoor to take the first set with two breaks of serve.
The 22-time grand slam champion was in no mood to waste any more time after a couple of short delays due to light rain as he attacked his Dutch opponent's serve early in the next set.
Firing winners at will from the baseline, the Serbian also showed soft hands with a vintage drop shot and followed it up with a stunning pass at the net before surging to a 4-0 lead.
Griekspoor managed to win a game and world No.1 Djokovic dropped serve with a double fault before sealing victory in style.
"It's been a great evening for me. Yesterday, I really had to work hard to get a win," Djokovic said.
"Tonight, right from the beginning I was sharp. I played better quality tennis. Maybe the last three or four games were not the best to close the match but I managed to find the good serve in the end.
"I'm very pleased... with the way I felt on the court and hopefully things can go in the right direction for tomorrow."
The Serbian will next face either Pavel Kotov or Hubert Hurkacz.
Earlier, holder Andrey Rublev saved five match points in a thrilling 1-6 7-6(6) 7-6(3) victory over Alejandro Davidovich Fokina.Â
The Russian will next take on Botic van de Zandschulp who beat Mikael Ymer 3-6 6-3 6-3.
"These matches, when it's over and somehow you win, it's always special. I don't know. I have no words to describe it," Rublev, who has saved match points en route to victory three times this season, said.
"One time maybe you feel lucky, you feel super tight and emotional because you almost lose. When it happens for a third time, you think now there's no chance.
"When you're three times lucky it's like, 'How many more Christmas presents do I have?'"
Lorenzo Sonego also advanced with a stunning 7-6(4) 6-4 win over fourth seed Felix Auger-Aliassime.
He meets Alexander Zverev after the seventh seed downed Australian Chris O'Connell 7-5 6-4 to reach the last eight of an event for the first time since his run to the Roland Garros semi-finals where he suffered a serious ankle injury.
