The Spaniard was far from his best, committing 15 unforced errors and three double faults in the decider, but capitalised on his lone match point courtesy of a Rublev double fault.
It was his 15th consecutive Masters 1000 win.
"Playing someone like Andrey, when you lose focus on two or three points, it can cost you the set or the match. I just stayed strong mentally and that's what I'm most proud of," Alcaraz said.
"It's just accepting the moment, accepting that I am playing a third set, that it's going to be a really tough battle, and I love that," he added.
Five-times grand slam champion Alcaraz, winner of 37 of his last 39 matches, advanced to his 12th Masters 1000 semi, equalling the mark of Italian top seed Jannik Sinner, who faces Frenchman Terence Atmane in the other last-four clash.
Alcaraz has set up a showdown with third-seeded Alexander Zverev, who dispatched fifth seed Ben Shelton in an hour and 17 minutes.
Zverev beat the American 6-2 6-2, dominating their match and sending down 12 aces in the process.
The German faced only one break point which he defended and then broke Shelton four times for his comfortable win.
Zverev's biggest concern on the night was his own health. He called for the doctor at 2-1 in the second set and appeared to be struggling with this breathing.
"Right now I'm not feeling too great… I'm not sure what happened," the 2021 Cincinnati champion said.
"I came out today and probably felt the best I've felt in a few months. Was feeling the ball incredibly well from both sides.
"In the first set I started feeling not so great and it got progressively worse. But I'm in the semi-finals and I'll do everything I can to be 100 per cent tomorrow."
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