Draper was heavy favourite to reach the final eight at the Foro Italico after a scintillating season that saw him clinch his first title at Indian Wells in March and finish as runner-up at the recent Madrid Open.
But after losing the opening set, Britain’s number one stormed back to overcome the Frenchman in two hours and 19 minutes at a sun-drenched SuperTennis Arena.
The 23-year-old will rise to world number four by reaching the semi-finals with Taylor Fritz crashing out to Marcus Giron in the earlier rounds.
But he will face an almighty challenge from Carlos Alcaraz after the Spaniard beat Karen Khachanov 3-6, 6-3, 7-5 in their last-16 match.
Former Australian Open and U.S. Open semi-finalist Khachanov started well and took a 3-1 lead before Alcaraz responded swiftly to level things up, making it 3-3 and sweeping the next three games to close out the opening set.
Despite being 3-2 down in the second set, Khachanov forced a decider — the first time the 28-year-old Russian had taken a set off Alcaraz in four meetings.
The Spaniard saw his lead slip away again after being up 4-1 in the third as Khachanov clawed his way back to draw level at 4-4, threatening an upset.
However, the 22-year-old Alcaraz steadied himself just in time, breaking Khachanov in the 12th game to clinch victory in two hours and 29 minutes.
"It feels amazing to get the win at the end against a really big and really tough guy like Khachanov," Alcaraz said.
"Physically I struggled a bit. Not any pain on any part of the body, but I was just tired. The match was really tough. I had to run a lot, so I'm just really proud of the way I fought for every ball."
Paolini wins rollercoaster
Jasmine Paolini became the first Italian woman to reach the Italian Open semi-finals since 2014 when she beat Russian world number 11 Diana Shnaider 6-7(1), 6-4, 6-2 on Tuesday.
In a stunning first-set collapse, Paolini was 4-0 up but lost the next five games before Shnaider then forced a tiebreak, which the 21-year-old Russian dominated.
Backed into a corner, world number five Paolini mounted a comeback of her own in the second set, winning six successive games after herself trailing 4-0.
Down a break in the decider, French Open runner-up Paolini continued to show resilience with the crowd behind her to became the first Italian woman to reach the last four of the WTA 1000 event since doubles partner Sara Errani just over a decade ago.