The 21-year-old played some stunning tennis in the closing stages of a dramatic 4-6, 7-6 (7/4), 2-6, 6-3, 6-3 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.
Cecchinato even overcame a point penalty served to him by the chair umpire in the fourth set, said to have left the court without proper permission after the third, per Christopher Clarey of the New York Times.
Cecchinato is the lowest-ranked French Open semifinalist in 19 years and the first Italian man to make it that far at any major in 40 years.
On court, Cédric Pioline, the former French star, started the post-match interview by reassuring Cecchinato, "You're not dreaming".
Yet the Serbian - who cruised past Rogerio Dutra Silva and Jaume Munar in the previous rounds - was on court for three hours and 48 minutes before finally recording a fourth successive win over Bautista Agut on clay.
The 12-time Grand Slam victor defeated Fernando Verdasco at Roland Garros.
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"He is one of the fittest guys on the tour so it was hard for him today", said Thiem, the only man to have defeated 10-time French Open champion Rafael Nadal on clay this year.
Putintseva led Keys 5-3 in their first set before a three-game swing saw the momentum shift, while 10th seed Stephens was more dominant in her run to beat Kasatkina for the second time in their three meetings. The last Italian man to take part in a major semi-final was Corrado Barazzutti at the 1978 French Open.
"To be able to win that first set and get ahead was crucial". His quarterfinal match with Marco Cecchinato may answer that question.
Since then, employing a smooth one-handed backhand, he has beaten players seeded No. 8 (David Goffin) and No. 10 (Pablo Carreno Busta), before adding Djokovic to his list.
Djokovic, Cecchinato said, "seemed unsure of himself at the beginning".
Djokovic won on his second match point with an easy smash at the net.
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. "Because if I played just to stay in points, I could never beat Novak".
Somehow, Cecchinato won the one that mattered most, the last.