The Bulgarian, a two-time Grand Slam semi-finalist, has still never reached the second week at Roland Garros in eight attempts.
Zverev is playing in a quarter-final at the Slams for the first time while Thiem has made the semi-finals in Paris for the last two years.
"He served for the match and I had to save lots of break points in the fourth and fifth sets".
In an extraordinary post-match press conference, Djokovic then said he may even skip Wimbledon where he has been champion three times.
"I'm just not thinking about tennis at the moment".
Two years ago, his career was nearly derailed when he became embroiled in match fixing allegations.
"Maybe I'm sleeping. It's fantastic, it's unbelievable for me".
In all honesty, not many were expecting a shock defeat at this stage.
"He played fantastic, so congratulations for a great performance", he said.
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Nothing should be taken away from Thiem's victory, however, after he carved an easier route to the quarters and is rewarded for his work with a spot in the final four, where he'll face Cecchinato.
The 12-time Grand Slam champion took the first two sets by a single break of serve and raced through the third as 34-year-old Verdasco ended with 48 unforced errors.
His hunger for the fight was clear for all to see too against Cecchinato in a thrilling tiebreak - the Serb roaring to the crowd, who chanted "Djoko Djoko" at crucial moments. And after, when I warmed up, it was better.
The Serbian tennis star, who made his return earlier this year after being sidelined for nearly six months due to an elbow injury, was involved in a tight battle against Spain's Roberto Bautista Agut on court Suzanne-Lenglen.
"I won my last match against Dominic Thiem, I think it was in the final of a Future (third tier) tournament", he joked.
Stephens, the 10th seed, had previously made the fourth round for four successive seasons and will next play either Australian Open champion and second seed Caroline Wozniacki or Russia's Daria Kasatkina.
Cecchinato is bidding to become the first Italian man to reach the semi-finals at a Grand Slam since Corrado Barazzutti in Paris in 1978.
It may be two years since Djokovic's last Grand Slam triumph, but Federer went nearly five years from Wimbledon in 2012 until the 2017 Australian Open before he added to his majors collection.
World number 98 Yulia Putintseva, a quarter-finalist two years ago, tackles Madison Keys of the United States.