It was not the routine victory Djokovic enjoyed in his return to action a day earlier against Tomas Etcheverry, as he recovered from a set down to eventually win in two hours 39 minutes.
The Serb took an early break in the first set but surprisingly cracked at 4-2 as his opponent broke back and then went on to win four consecutive games to claim the opener 6-4.
The pair were inseparable in the second set as neither could force a break to gain the advantage, with the encounter rumbling inevitably towards a tiebreaker.
Djokovic showed his steel and composure to level the match with an imperious performance in the tiebreak, claiming it with minimal fuss.
The top seed's experience was in evidence in the fifth game of the decider when he capitalised on some nervous serving from the Dutch 27-year-old to break.
But Griekspoor grittily kept his tournament hopes alive and was rewarded when he broke his opponent's serve at 4-3, with Djokovic sarcastically applauding the Paris crowd afterwards.
This only fired up the Serb and he responded immediately, taking the very next game on the Dutchman's serve to love, before sealing the match with his next service game.
The 36-year-old moves on to face either Daniel Altmaier or Holger Rune in the quarter-finals on Friday.
Fourth seed Jannik Sinner withdrew before his third-round match, blaming "fatigue" after finishing his previous match after half-past two in the morning.
The night before, the Italian started his match against American Mackenzie McDonald after midnight and battled to a three-set victory in two hours 16 minutes.
Sinner was due on centre court on Thursday as the fourth match on the schedule to face Australian 13th seed Alex de Minaur.
'Zero care for welfare'
Darren Cahill, who coaches Sinner, said on Instagram that he was happy with the victory but criticised "zero care for players' welfare with Paris scheduling".
Casper Ruud, the eighth seed, who lost to Francisco Cerundulo on Wednesday, echoed the criticism on Thursday.
"Bravo atptour way to help one of the best players in the world recover and be as ready as possible when he finished his previous match at 2:37 am this morning 14.5 hours to recover.. what a joke," the Norwegian posted on X, formerly Twitter.
Sinner arrived in Paris from Vienna where he won the title on Sunday in a final against Daniil Medvedev that lasted more than three hours.
Stefanos Tsitsipas advanced to the quarter-finals and also locked up his place in the season-ending eight-man ATP Finals as he beat Alexander Zverev 7-6 (7/2), 6-4.
Tsitsipas became the sixth player officially qualified for the November 12-19 event in Turin after his first win this season in eight matches against top-10 players.
"It's awesome. What a great relief to finally get that spot," said Tsitsipas.
"I've been working extremely hard the whole year to be in Turin, and it's a moment of relief that I've made it to one of my favourite events of the year."
Zverev still holds the seventh spot in Turin but the two men immediately below him, Rune and Hubert Hurkacz are still going in Paris and could collect enough points from the tournament to overtake Zverev.
In the quarter-finals, Tsitsipas will face Russian Karen Khachanov, the winner in Paris in 2018. Khachanov, the 16th seed, beat compatriot Roman Safiullin, who had eliminated Carlos Alcaraz, 4-6, 6-4, 6-2.
Hurkacz advanced with a 6-4, 6-3 win over unseeded Argentine Cerundolo. The Pole will face Bulgarian Grigor Dimitrov in the last eight.
Dimitrov opened the day's action by racing to a straightforward 6-2, 6-2 victory over Alexander Bublik of Kazakhstan.
Dimitrov, the world number 17, showed little sign of any effects from his three-hour win against third seed Medvedev the previous afternoon as he dispatched Bublik in just over an hour.
Andrey Rublev also moved into the last eight, beating Botic van de Zandschulp of the Netherlands 6-3, 6-3.
The Russian has already qualified for the ATP Finals in Turin later this month. He will face the well-rested De Minaur in the Bercy quarter-finals.
