Atmane followed up his victory over fourth-ranked Taylor Fritz with a 6-2, 6-3 win over ninth-ranked Rune.
The 23-year-old will now face the ultimate test against defending champion Sinner, who said he "felt great" as he thrashed Felix Auger-Aliassime 6-0, 6-2.
"I don't think any words can describe how I feel right now," the 136th-ranked Atmane said after blasting 22 winners past Rune, 14 of them off his powerful lefty forehand.
"It's pretty insane to be honest," added Atmane, who will move inside the top 100 in the rankings for the first time.
"I cannot believe it. Being here in the semi-finals of a Masters 1000, breaking into the top 100. It means a lot to me."
He will face a formidable challenge in Wimbledon champion Sinner, who powered through a rapid-fire opening set and bounced back quickly after dropping serve to open the second.
Sinner prevailed in a brief 71 minutes as he claimed an eighth victory in a row at this event and his 30th match win this season.
"I felt great on the court today, you could see that," said the Italian, who turns 24 on Saturday. "But every day can be different. Let's see what I can do in the semis."
Sinner took advantage of Auger-Aliassime's serving troubles to take the first set in less than 30 minutes, the Canadian delivering three double-faults in the final game.
After going down a quick break in the second Sinner broke back to level at 2-2 and advanced as Auger-Aliassime coughed up his eighth double-fault on match point.
"I served well today, that was the key for me," Sinner said. "I had a small drop in the second set and I'm happy that I broke back. He moves and serves well, he's difficult to play."
In the day's only other men's match, fifth-seeded Ben Shelton - coming off a title in Toronto - posted a 6-4, 6-4 victory over Jiri Lehecka to book a quarter-final clash with third-seeded Alexander Zverev.