Rublev, who has now reached the quarters eight times at the Grand Slams but has never made the last four, was seemingly cruising into the last eight when he took the first two sets on Centre Court.
Yet his Kazakh opponent somehow conjured a way back into the contest and ensured it went the distance before seventh seed Rublev wrapped up a 7-5 6-3 6-7(6) 6-7(5) 6-4 victory.
Had the Russian lost it would have been a remarkably bitter pill to swallow, having been two points away from victory in the third set tiebreak, passed up two match points in the fourth and led 5-3 in the second breaker only to lose four points in a row.
Having secured a decisive break to go 5-3 up in the fifth, Rublev pulled off one of the most remarkable shots of the tournament so far with a diving forehand winner to bring up match point and then banged down an ace to finish the contest.
"Probably it was the most lucky shot ever," he said in his on-court interview. "I don't think I can do it (the shot) one more time."
Rublev could be known as Mr Quarter-Final at the Grand Slams and his victory ensured he will now have a full set of last-eight appearances at the majors, having never reached that stage at Wimbledon before.
Whether he can go one step further at a major for the first time is a feat few will fancy him achieving at Wimbledon, with seven-time champion Novak Djokovic his likely next opponent, providing the Serb overcomes Hubert Hurkacz later on Sunday.