This victory marked his fourth appearance in the US Open semi-finals, setting the stage for a potential showdown with the defending champion, Carlos Alcaraz.
Medvedev, seeded third and the 2021 US Open champion, displayed remarkable resilience as he battled back from a deficit in all three sets.
He handed Rublev his ninth consecutive Grand Slam quarter-final defeat, leaving the frustrated opponent in disappointment.
Medvedev’s next opponent in the semi-finals will be determined from a match between Alcaraz and the German 12th seed, Alexander Zverev.
The match against Rublev proved to be physically demanding, with the temperature soaring to 95 degrees Fahrenheit (35 degrees Celsius) in New York.
In the midst of the intense contest, during the third set, Medvedev was overheard muttering to a nearby TV camera, “one player is going to die, and they’re going to see,” as he grabbed a towel in the middle of the game.
“The conditions were brutal. The only good thing is that both players suffer, so it’s tough for both of us,” Medvedev said after the 2hr 48min encounter, during which the extreme heat policy was in effect.
“At the end of the first set I kind of couldn’t see the ball anymore. I kind of just played with sensations.”
Medvedev overturned a 3-0 defeat in the opening set in a repeat of their 2020 quarter-final here.
He dropped serve twice in the second set but hit back for a 4-3 lead before summoning the trainer complaining of breathing difficulties.
Medvedev resisted three break points the very next game and broke once more to claim a two-set lead.
He then used the time between sets to cool down off court.
“I couldn’t wait to go to the cold shower, but the thing about that is when you come out either you can’t move because your body blocks or you feel better,” said Medvedev.
“I just thought I don’t care, I’m just going to do it, but I didn’t need to, so I am going to do it now.”
Rublev twice surged in front in the third set only for Medvedev to respond on both occasions, sealing victory in a marathon final game courtesy of his nine break of the match.
(With inputs from AFP)