Both players were bidding to reach the last-eight stage at a major for the first time in their careers but it was Zverev, who recorded a 4-6 7-6 (7-4) 2-6 6-3 6-3 win in three hours and 29 minutes.
These hurdles have finally been overcome, but Zverev has had to dig deep. He missed an easy forehand on his first match point, biting down on his fingers in frustration.
Italian Cecchinato won in four sets, scrambling over the line in an epic tie-break on his fourth match point.
Thiem beat No. 19 Kei Nishikori of Japan 6-2, 6-0, 5-7, 6-4 on Court Philippe Chatrier.
Thiem showed no mercy as he continued to dominate, racing into a 4-0 lead in a flash in the third set by mixing up his game expertly.
Thiem had defeated Zverev in four of their six previous meetings on tour, however, the Austrian was seeking revenge after losing to his opponent in the final of the Madrid Masters last month.
"I don't know if I'm going to play on grass", Djokovic, who won the last of his 12 Grand Slam titles in Paris two years ago, told reporters crowded into a small interview room after he declined the opportunity to use a much larger one.More news: Stephens stands in Halep's path
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In the fourth set, too, Nishikori fought on, benefitting from Thiem's unforced errors while looking visibly frustrated at being unable to stay in rallies.
Marco Cecchinato beat former champion and 12-time major victor Novak Djokovic 6-3, 7-6, 1-6, 7-6 to become the first Italian man to reach a Grand Slam semi-final in 40 years.
"I knew I'm not going to win the match". It's the best, a five-set match.
But marathon woman Caroline Wozniacki - she finished the New York City Marathon in 2014 - trailed Daria Kasatkina 7-6 (7-5) 3-3 when darkness halted proceedings at around 9:20 p.m. Paris time. - Williams, discussing the chest muscle injury that caused her to pull out of the French Open before she was supposed to face Sharapova.
The world number three said: "I thought it was a bit ridiculous, to be honest, because I was on the other side of the court".
Italy's Marco Cecchinato celebrates winning his fourth round match against Belgium's David Goffin in four sets 7-5, 4-6, 6-0, 6-3.
Madison Keys reached her first French Open semifinal by defeating unseeded Yulia Putintseva 7-6, 6-4 - and her reward is a showdown with compatriot Sloane Stephens.
After squandering two break-point chances, Zverev made amends in the Russian's following service game in the second set to take command but Khachanov hit back on his fifth break point when the German served for the second set to secure the break back.
Asked whether he could have envisioned, even as recently as April, that he would get this far at a major tournament, Cecchinato answered with one word, "No", before breaking into as wide a smile as can be.