But from somewhere Cecchinato got a second wind and took the fourth in a thrilling tie-break, converting a fourth match point to win 6-3 7-6 (7/4) 1-6 7/6 (13-11).
Novak Djokovic served for the fourth set at 5-3 but got broken.
Analyzing the match, Djokovic added: 'He played fantastic and credit to him. "It was the first time I have won on this court and I hope there are many more to come".
"It was an awesome, high level match", said Zverev.
'I need to - I don't know.
US Open champion Sloane Stephens cruised past Estonian Anett Kontaveit 6-0, 6-2 to book a last-eight spot for the first time.
Two years ago, his career was nearly derailed when he became embroiled in match fixing allegations.
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Clinton also said that he paid dearly for the scandal , stating he faced millions in debt after he left the White House . The House Republicans' move against Mr Clinton in 1998 made him only the second U.S. president to have been impeached.
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A fractured finger has forced Eoin Morgan out of the Hurricane Relief Twenty20 worldwide against West Indies at Lord's tommorrow. Other names in the World XI squad include Rashid Khan, Adil Rashid and India wicket-keeper Dinesh Karthik .
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They should have contacted her personally "2 talk rather than do it in a public arena", she tweeted this afternoon. He scoffed & said: "what u have done is egregious, and unforgivable.' I begged 4 my crews jobs".
Cecchinato, who was suspended for match-fixing two years ago, before his ban was overturned on appeal, arrived in Paris having not won a single Grand Slam match.
Cecchinato said he was stunned by his triumph, coming just two years after his career was nearly derailed when he became embroiled in match fixing allegations.
Men's second seed Alexander Zverev survived a scare as he had to save a match point before beating Damir Dzumhur on Friday to reach the fourth round of the French Open.
In a tense encounter, both men were slapped with code violations for coaching before Cecchinato gallantly saved three set points.
2016 champion Djokovic, in a 12th Roland Garros quarter-final and 40th at the majors, is bidding to reach his 32nd Grand Slam semi-final when he faces world number 72 Cecchinato. 'Obviously I really expected to play better on clay.
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.
Zverev, considered among a select few capable of dethroning defending champion and French Open king Rafael Nadal, had to do it the hard way as he was on the right side of a 6-2, 3-6, 4-6, 7-6, 7-5 scoreline.
From that point there was no way back for the Spaniard, who was aiming to reach his first Roland Garros quarter-final at the 15th attempt.
"First week, my goal is to get to the second week", said Keys.