Cecchinato had never won a match at a Grand Slam event before this year's tournament at Roland Garros.
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Zverev produced some stunning tennis in the closing stages of a dramatic 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.
Dzumhur, the world number 29, was in the middle of a tough five-set match at the French Open, against second seed Alexander Zverev.
But while Dzumhur - who has never been beyond the third round of a major - may have felt the nerves, the 21-year-old German played some big points at crucial moments - in a match one suspects he would have lost in recent times. He missed an easy forehand on his first match point, biting down on his fingers in frustration. Though the world No. 3-ranked Zverev has won eight career titles-including three ATP Masters 1000 crowns-he has yet to find success at the Grand Slam level.
At 6ft 6in, he has been touted as a future world number one, but glimpses of his potential were interspersed with a failure to close out vital points and a hint of fatigue.
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But Zverev fought back from 2-1 down to win the final two sets and become the youngest quarter-finalist in Paris since a 20-year-old Juan Martin del Potro in 2009.
"Me and my brother (Mischa) are guys that spend three, four hours a day in the gym, lifting heavy weights, on the treadmill". "I'm very happy about being in the quarterfinals here going the hard way, going the long distance. showing everybody that I can play for as long as I need to".
Alexander Zverev jokes that it took him five sets to beat Karen Khachanov because he 'might as well stay longer for practice!' But he then picked himself up with Dzumhur's help.
It was a brief reprieve as Khachanov's aggressive volleying tactics earned him a third set-point chance and this time the Russian took it to clinch the opener.
"It's the best, a five-set match".
Novak Djokovic equaled Roger Federer's mark of reaching nine consecutive quarterfinals, by beating No. 30 seed Fernando Verdasco 6-3, 6-4, 6-2.