The Serbian No. 1 clawed his way back into the contest and had three set point chances at 6-5 on Cecchinato's serve but was denied.
Paris: World number 72 Marco Cecchinato became the first Italian man in 40 years to reach a Grand Slam semi-final on Tuesday with a breathtaking 6-3, 7-6 (4), 1-6, 7-6 (11) victory over 12-time Major victor Novak Djokovic.
Djokovic won the tournament in 2011, 2014 and 2015 but has failed to make it beyond the quarter-finals in the past two years.
He led Cecchinato 5-2 in the fourth set, but lost the lead.
In an extraordinary post-match press conference, Djokovic then said he may even skip Wimbledon where he has been champion three times.
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Rose, four adrift going into the final round, had two birdies in his first three holes before dropping two shots on the back nine. Woods seems to have found his new old (or old new?) groove, and his play this week is the new expectation going forward.
He replied: "I don't know". I just came from the court.
"I had a lot of courage, especially toward the end of the tiebreaker", Cecchinato said.
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. "It's unbelievable", he said.
He would not face 10-time champion Rafael Nadal - a rematch of the 2012 and 2014 finals - until the final.
In the women's draw, Madison Keys is into the last four at Roland Garros for the first time in her career after seeing off the challenge of the unseeded Yulia Putintseva.
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Commerce Department numbers show, the United States recorded a trade surplus with Canada for each of the past three years. The World Trade Organisation's former chief, Pascal Lamy, also said the damage would likely be limited in concrete terms.
Djokovic, who came to Paris with his lowest seeding since 2006 after a hard past 12 months in which he has dealt with an elbow injury, had begun to look like his old self at Roland Garros - dropping just one set en route to the last eight. I don't know what I'm going to do.
The 31-year-old Serb said he did not "want to talk about tennis right now".
The 10th seed produced a clinical performance to down Kasatkina 6-3, 6-1 on Court Philippe Chatrier, firing 17 winners.
An elbow injury has limited the 12-time Grand Slam champion's playing time this year, while he was treated on his neck during the defeat by Cecchinato.
Keys struck 30 winners past a battling Putintseva in an impressive display as she continued her recent revival on clay, having never reached the quarter-finals in Paris before this week.
"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.
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Republican Phil Scott called on lawmakers to look for ways to attract and retain more young people. You will also need to prove you are a full-time remote worker for a company outside of the state.
"I think it's changed my life. It feels completely different here".