The Bulgarian won the ATP Tour Finals in London a year ago but his wait for a grand slam title will go on after a 7-6 (7-4) 6-2 6-4 loss.
An elbow injury has limited the 12-time Grand Slam champion's playing time this year, while he was treated on his neck during the defeat by Cecchinato.
Cecchinato (it's pronounced cheh-key-NAH'-toe) came through on his fourth match point, looping in a backhand return victor as Djokovic tried to surprise him with a serve-and-volley attempt.
Djokovic had also squandered three set points in the second set and failed to serve out the fourth in the ninth game after being 5-2 in front.
"I don't know if I'm going to play on grass", he said.
"I just could never turn it around out here".
"At the beginning of the year when he lost a match he wasn't as sad as he was today".
"It was an wonderful, high-level match", said Zverev. He arrived at Roland Garros with a 10-7 record this season. "It's awesome", he said.
"It's awesome", said the 25-year-old who is the first Italian to get this far at a Slam since Corrado Barazzutti at Roland Garros in 1978.
Novak Djokovic advanced to the fourth round of the French Open on Friday by beating Roberto Bautista Agut 6-4, 6-7 (6), 7-6 (4), 6-2.
Like her compatriot and good friend Keys, US Open champion Stephens is in the quarter-finals for the first time and hoping to set up a repeat of last year's final in NY. This will be Zverev's first quarterfinal match in a Grand Slam, so Thiem edges him in experience.
"Is good for my opponent in Wimbledon", he said. 'Obviously I really expected to play better on clay.
"I know how to handle a Grand Slam now, how... to get that deep in such a tournament, and I think everything gets better with experience".
US Open champion Sloane Stephens cruised past Estonian Anett Kontaveit 6-0, 6-2 to book a last-eight spot for the first time.
The 10th seed produced a clinical performance to down Kasatkina 6-3, 6-1 on Court Philippe Chatrier, firing 17 winners.
'How? I mean, we'll figure.
The Serb raised his level to win the third 6-1 in just 27 minutes as he clawed his way back into the quarter-final and then called for the trainer again after reducing his deficit, this time to receive treatment on his right leg.
From that point there was no way back for the Spaniard, who was aiming to reach his first Roland Garros quarter-final at the 15th attempt.