
survived a three-set battle with Corentin Moutet - and another double-bounce drama - to reach the Rome Masters quarter-finals and set up a fascinating contest with Carlos Alcaraz.
And the British No.1 will become the new world No.4 if he reaches the last four by beating Alcaraz in his final event before the French Open.
Madrid Masters finalist Draper recovered from a poor first set in the Italian capital to beat his fellow left-hander 1-6 6-4 6-3 in two hours and 20 minutes.
The world No.5 was not on top form but he showed grit and mental strength to fight his way to victory.
Marathon man Moutet had beaten No.9 seed Holger Rune in the last round in three hours and 44 minutes - the longest match here since 1991 - but suffered a hamstring injury in the final set.
Draper, seeking his first ever claycourt title, had reached the final at the Madrid Masters in his last event but had smashed his racket in frustration before beating Czech qualifier Vit Kopriva in the last round.
And he made 15 unforced errors and was broken twice in a ragged 38-minute first set.
But the match's big talking point came at 3-3 in the second set on the Draper serve. On the first point, the British No.1 stretched to get to a Moutet drop shot on the backhand and got the ball over the net. The Frenchman hit his short forehand return into the net but then immediately complained to the umpire that the ball had bounced twice on Draper's side of the court. The pair endured a brief tense exchange at the net after the drama.
Moutet demanded a VAR review but German umpire Nico Helmwerth correctly ruled that Draper had just got his racket under the ball before the double bounce. Sky Sports analyst Miles Maclagan said: "For me there was no doubt. He got the ball before it bounced twice."
Speaking about the double bounce controversy, Draper said: "I believe I got the ball up. I couldn't really see it on the screen too well. I thought I got it up.
"He said I hit the ball into the ground but on the review - and the ref looked at it closely - it looked like the ball went up. I don't know - it felt like I got it up."
Draper was involved in a similar controversy at the Cincinnati Masters last August when he won match point against Felix Auger Aliassime after appearing to scoop the ball up after a double bounce. The British No.1 admitted he had been hurt by all the subsequent criticism and called for VAR to be available for big calls.
The world No.5 held his serve before a distracted Moutet lost his serve to love - including two double faults - to allow Draper to serve for the second set at 5-3. But he was broken to love before breaking Moutet again when the Frenchman set a forehand lob long on the first set point. Draper cupped his hand to his ears to milk the applause while Moutet threw his racket to the court in frustration.
At 3-3 in the decider, Moutet suffered an injury to his left hamstring and took a three-minute medical timeout for treatment at the end of the game.
And the struggling Frenchman again tossed his racket to the court as he gave up two break points in the following game - and Draper took the second to serve for the match. And he took his first match point with a backhand volley winner.
Draper can overtake Taylor Fritz to become world No.4 after the American lost in the second round here.
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