Cecchinato (it's pronounced cheh-key-NAH'-toe) came through on his fourth match point, looping in a backhand return victor as Djokovic tried to surprise him with a serve-and-volley attempt.
Keys, the US Open runner-up to Stephens a year ago, will face Kazakhstan's Yulia Putintseva for a place in the semi-finals, after the world number 98 saw off Barbora Strycova 6-4, 6-3.
"I think that's going to be the number one priority for me now, to kind of step out from the tennis for a little bit, try to watch some matches and sort of try to progress somehow and just get better".
"First week, my goal is to get to the second week", said 23-year-old Keys.
Cecchinato won on his fourth match point by hitting a backhand victor down the line as Djokovic tried to serve and volley.
After a tight opener was settled by a tiebreak, Verdasco steamed ahead and Dimitrov simply had no answers as he fell 6-7 (4-7) 2-6 4-6 in a match lasting just under two-and-a-half hours - something he suggested has always been the case when playing in the French capital.
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Djokovic sprinted out to 5-2 but was reeled back in to 5-5, having squandered three points for a double break in the sixth game.
The Bulgarian won the ATP Tour Finals in London previous year but his wait for a grand slam title will go on after a 7-6 (7-4) 6-2 6-4 loss.
"Is good for my opponent in Wimbledon", he said.
German second seed Zverev simply ran out of gas, paying a heavy price for needing three successive five-set matches to reach his first Grand Slam quarter-final.
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Seeded 20th after a well-documented slump, Djokovic blunted Verdasco's explosive firepower with his signature counter-punching to book an unlikely last-eight date with Marco Cecchinato.
Cecchinato and Djokovic know each other well: They have practiced together in Monte Carlo, where Cecchinato trains at a tennis academy and Djokovic has a home.
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"I definitely thought about it, but I didn't want to pull out for the first time of my career in a Grand Slam quarter-final", said Zverev.
"I hope we have many more matches at the Grand Slams but when we are 100 percent".
The clay courts of the French Open have never been a strong suit for Dimitrov, who was knocked out of the third round for a third time in his career.
"I'm a better player in general, for sure".
And the one that won deserved to win the match, and that was Marco today.
A drop of serve when in position to win the match did not matter for Djokovic, though, as Bautista Agut was broken in the next game, sending him through to a tie with Verdasco.
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The FIFA World Cup is scheduled to begin on June 14, as 32 teams battle for honors in the World's biggest football tournament. "For his part, Mohamed Abu El-Ela said the period of absence of Salah due to injury will not exceed three weeks".
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