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."
For Rune, the defeat was all the more disappointing because he had reached the semi-finals of the tournament last year. As a result, he will lose 300 ATP points, which will see him plummet to 9th or 10th place in the ATP rankings before the start of the US Open.