Following an early exchange of breaks on a blustery day, Auger-Aliassime pounced on Cobolli's serve in the 10th game to claim the opening set on Wednesday, after which the roof on Court Philippe Chatrier was closed.
Cobolli, one of three Italian men still in the men's draw after Jannik Sinner's premature exit, battled back from 1-3 down in the second set with the playing conditions suddenly improving.
"I think we played two different matches today," Cobolli said in an on-court interview. "In the first set it was very windy and tough to play.
"I went to the toilet to think a bit. I tried to change something. I thought 'this is the best court I've played on in my life, because I can show my best tennis'. I said to myself to fight, this is the chance of my life and I must give everything in all my matches. So today I'm really happy.
"Tennis is like this, it changes every day, this week changed a lot and we have to be prepared, accept everything. I did it today and that's why I won this match."
Having levelled the contest at one set apiece, the 10th seed dialled up the intensity in the seventh game of the next set and put himself just one set away from his maiden Grand Slam semi-final appearance.
The 24-year-old thumped his chest when he edged ahead in the fourth set, and he went on to comfortably secure the victory.
Cobolli said he was a superstitious person and would not change his routine to watch that match from the venue.
"The tension of my racket has been the same since two weeks, even with the warm weather," Cobolli added, explaining how steadfast he is with his routines during tournaments.
"I'll do my same routine today as every day, have dinner with friends. I don't care which Matteo is in the semi-finals. It's been the best week of my life but I'm still missing something, so I have to fight again."
Cobolli will now face Matteo Arnaldi, who came out on top in an all-Italian clash to reach his maiden Grand Slam semi-final after opponent Matteo Berrettini retired injured while trailing 7-5 5-2.
Berrettini, who needed a medical time out for what appeared to be hip pain after three games in the second set, threw in the towel at the change of ends after being encouraged to do so by his entourage.
Arnaldi said: "It's a tough one. We both played a lot so it's normal to not be at our best but you never wish someone's tournament to end like this.
"We're doing such a good job in Italy. I hope he recovers, the grass season starts soon and he's someone who is going to be tough to play against."
