Kei Nishikori 6-2, 6-0, 5-7, 6-4.
But all eyes will be on Monday's showdown between Serena Williams and Maria Sharapova, with the latter trying to end an 18-match losing streak to her bitter rival. I just came from the court. "Maybe it's a dream, because now I'm very, very happy for this match", said Cecchinato.
Fourth seed Dimitrov was beaten 7-6 (7-4) 6-2 6-4 by Fernando Verdasco on Court 1 at Roland Garros, two days after being pushed to five sets by world number 57 Jared Donaldson.
Cecchinato, who will face No. 7 seed Dominic Thiem in the semifinals, is unlikely to match that story line, but he will always have Tuesday's stirring victory over Djokovic, the 2016 French Open champion and a former No. 1.
"He held his nerves amazingly well in important moments", acknowledged Djokovic, who said he isn't certain whether he will play at Wimbledon.
US Open champion Stephens brushed aside Russian Daria Kasatkina to set up a repeat of last year's Flushing Meadows final with fellow American and good friend Keys.
Cecchinato struggled to comprehend his triumph, coming just two years after his career was nearly derailed when he became embroiled in match-fixing allegations.
"I definitely need to take some time off now to kind of reassess the whole clay-court season, to be honest", said Dimitrov. For me to beat Djokovic in a quarter-final at Roland Garros it's awesome.
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.
"I'm a better player in general, for sure".
Earlier in the day Keys, the 13th seed, reached the semi-finals for the first time by seeing off Kazakhstan's Yulia Putintseva 7-6 (7/5), 6-4.
Craig O'Shannessy, the strategist for the ATP Tour and an adviser to Djokovic, was impressed with the Italian's game plan, suggesting he could even threaten Nadal, should they both make the final.
The 25-year-old, who was ranked 957 just six weeks before her maiden Grand Slam triumph last year after a series of injury problems, is now into the last four of a major for the third time.
"To be able to win that first set and get ahead was crucial". You know, for me, not having so many matches in the last period, this is great.