The Spaniard seized total control of the match in the second set, as he held serve, despite facing two break points, and made the most of two out of three break points to double the lead.
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.
As his backhand return on his fourth match point landed beyond the outstretched arm of Djokovic, the tears flowed.
The 31-year-old also needed a medical timeout to treat a neck injury which looked as if it would derail him when he slipped 0-2 down in the second set to the Italian outsider.
Djokovic claimed a close opening set, clinching the first break in the match in the 10th game to move ahead.
"It's a pity I could not capitalize on the chances I had", Djokovic said.
Spain's Fernando Verdasco awaits in the fourth round after he stunned Bulgarian fourth seed and Association of Tennis Professionals World Tour Finals champion Grigor Dmitrov with a 7-6, 6-2, 6-4 win. He arrived at Roland Garros with a 10-7 record this season.
Djokovic sprinted out to 5-2 but was reeled back in to 5-5, having squandered three points for a double break in the sixth game.
But after his defeat in a French Open quarterfinal on Tuesday, Djokovic was distraught-and unclear about where and when he would play next.
The 23-year-old American has now reached at least the last-eight of all four majors, after ending the dream run of Buzarnescu, who had never won a Grand Slam match before the tournament.
After a good start where he reached the semi-finals in Monte-Carlo and the last-eight in Barcelona, he lost back-to-back matches in Madrid and Rome.
"Obviously. making the fourth round four times or whatever is pretty good, but to finally get over that hump of quarter-finals feels very nice", she said.
Cecchinato and Djokovic know each other well: They have practiced together in Monte Carlo, where Cecchinato trains at a tennis academy and Djokovic has a home.
"I obviously rely on what I learned there and how to manage my emotions and manage the moment, but there were so many late nights and I was so exhausted".
"I definitely thought about it, but I didn't want to pull out for the first time of my career in a Grand Slam quarter-final", said Zverev. "Now I beat David Goffin, (who) is one of the best players in the world".
"Sometimes you have to battle to get the win and that's what I did today".