Italian Cecchinato won in four sets, scrambling over the line in an epic tiebreak on his fourth match point.
The storm seemed to have blown itself out and Djokovic served for the fourth set at 5-3.
It was a classic, Djokovic wasting three set points and Cecchinato unable to take the first of three for the match.
Meanwhile, another potential contender for the French Open title, German Alexander Zverev survived a scare from Damir Dzumhur before finally coming through with a 6-2, 3-6, 4-6, 7-6 (7-3), 7-5 victory, spending nearly four hours on court. What can I say?
Seventh seed Dominic Thiem muscled his way into the fourth round of the French Open, putting out Italian claycourt specialist Marco Berrettini 6-3 6-7 (5) 6-3 6-2 to set up a tie with Nishikori.
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To achieve that feat, Cecchinato, who in 2016 was suspended for 18 months for match fixing before his ban was lifted following a successful appeal, displayed an impressive range of claycourt tennis skills.
"I don't know. I don't know what I'm going to do".
Cecchinato struggled to comprehend his triumph, coming just two years after his career was nearly derailed when he became embroiled in match-fixing allegations.
"Is good for my opponent in Wimbledon", he said.
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In a tense encounter, both men were slapped with code violations for coaching before Cecchinato gallantly saved three set points.
The Russian also reached the Indian Wells final earlier this year, and has now won six times against players ranked in the world's top two.
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.
Thiem boasts being the only player to have defeated Rafael Nadal on clay prior to Roland Garros this year in the Madrid quarter-finals, ending the 10-time French Open champion's 21-match winning streak on clay.
Keys had always struggled on clay before this year, but has now reached the last eight at all four Grand Slams after impressively cruising through the draw without dropping a set.
The 23-year-old Svitolina, who trains at Roland Garros and was a quarter-finalist in 2015 and 2017, had been tipped as one of the potential winners for the second year in a row after winning the Italian Open, but she never found her groove.
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"Honestly, the (US) Open feels like it was 12 years ago at this point", said Keys, who is yet to drop a set in the tournament.