Canada's Auger-Aliassime produced a flurry of aces to take the first two sets and force the world number two to do it the hard way, but Medvedev's fighting spirit came to the fore as the match went on.
After saving a match point in the fourth set, Medvedev came back from two sets down to set up a deciding fifth set, where a long return from the Canadian saw Medvedev complete his storming fightback.
The US Open champion took the contest 6-7 (4-7) 3-6 7-6 (7-2) 7-5 6-4 to advance to the final four, and he credited inspiration from absent champion Novak Djokovic for his thrilling win.
Speaking during his post-match interview on the court, Medvedev said: "I was not playing my best and Felix was playing unbelievable, serving unbelievable. I told myself, what would Novak do?
"He and Rafa and Roger have won so many matches like this - if he wants to win it he has to fight for it. I managed to raise my level during the game, when they closed the roof I felt the momentum in my favour.
"I was sweating like hell and made a few double faults because my hand was so slippery. The roof definitely helped me to be less sweaty."
Looking ahead to the next round, Medvedev added: "It's the semi-final of a Grand Slam, even if I'm in the worst shape of my life I'm going to try to make it tough for my opponent."
Despite posting fewer aces, more double faults and a lower win % on his first serves, Medvedev ultimately advanced to the final four, where he will face Stefanos Tsitsipas after the Greek downed Jannik Sinner in straight sets. body check tags ::