Cecchinato cruised through the opening set as Djokovic was suffering from neck pain before the Serbian fought back after the loss of the second.
In an epic fourth set tiebreaker, Djokovic saved three match points but wasted three set points as Cecchinato became the lowest-ranked man in the semi-finals in Paris since 100th-ranked Andrei Medvedev in 1999.
"It's wonderful", 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.
Against Djokovic, his sliding across the court was almost flawless, as was his ability to disguise his angles until the last moment in what was perhaps the match of the tournament so far.
When pressed on whether that meant he would not play at Wimbledon, the 31-year-old Serb was non-committal.
Djokovic was back with his former coach, Marián Vajda, who said Djokovic was not at his best at Roland Garros, but improving.
The 20th seed reached the last 16 - where he will face Fernando Verdasco - at Roland Garros for the 12th time in his career after triumphing 6-4 6-7 (6-8) 7-6 (7-4) 6-2 on Suzanne Lenglen.
Zverev produced some stunning tennis in the closing stages of a dramatic fourth-round victory and will next face Dominic Thiem who reached a third successive quarter-final in Paris by seeing off Japan's Kei Nishikori 6-2, 6-0, 5-7, 6-4.
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He won the last of his 12 Grand Slam titles at Roland Garros two years ago.
Thiem, who lost in the semis to Djokovic in 2016 and Nadal past year, said he is now better-equipped to go all the way.
Verdasco, world No. 35, coped with world No. 5 Dimitrov's winners, controlling the serve points, only conceding his service game three times out of ten opportunities.
"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". "So now it's really just managing your expectations and your nerves and the moment".
Serbia's Novak Djokovic with Spain's Fernando Verdasco after winning their fourth round match.
Even as he conceded the second set, Djokovic managed to regain his composure as he went on to pull off a 6-4 6-7 7-6 6-2 win over Agut in an intriguing third-round battle.
Two years ago, his career was nearly derailed when he became embroiled in match fixing allegations.
Cecchinato, who had never won a Grand Slam match in his career before Roland Garros, will take on Austrian Dominic Thiem for a place in Sunday's final.