Zverev, a semi-finalist for the last three years, had not dropped a set in his opening two matches, including a first-round win over Rafael Nadal, but was pushed all the way by the Dutch 26th seed.
"Wow, incredible match, incredible player, he's unbelievably dangerous, I always struggle against him," a relieved Zverev said.
"I don't like to play against him, but all the credit to him, taking it all the way. An incredible fighter to the end."
Zverev was on the back foot when Griekspoor broke early and raced into a 3-0 lead before comfortably winning the opening set, but the second turned into a real battle.
Zverev cut a frustrated figure as he faced a break point at 1-1 in a game which dragged on for five deuces and almost 11 minutes, but he held on and finally broke serve at 5-4 up to level the match.
The German made lighter work of the third set and when Griekspoor took a medical timeout for treatment on his leg it appeared Zverev would have little trouble closing out victory.
The Dutchman forced a deciding set, however, and held a 4-1 advantage after two breaks of serve before Zverev roared back to win four consecutive games.
Griekspoor kept his nerve to take it to a tiebreak but, after an enthralling match lasting over four hours on Court Philippe Chatrier, it was Zverev who fired down an ace to clinch the victory.
"To be honest the atmosphere today was unbelievable, I loved every single second of it," Zverev said.
"The support that I get here at this Grand Slam I don't think I get at any other Grand Slam so I really appreciate it."
Zverev, now on a nine-match winning streak, having claimed the Italian Open title, will play the winner of the rain-interrupted third-round match between Danish 13th seed Holger Rune and Slovakia's Jozef Kovalik.
