The Russian, a Qatar winner in 2020, had to dig deep to down the Australian 6-1, 3-6, 7-6 (10-8).
Fifth-seeded Rublev let slip a 5-2 lead in the final set as De Minaur scrambled back into contention to force a tiebreak.
Rublev finally got over the line, the 27-year-old saving a match point himself in a memorable quarter-final clash in Doha.
"When I had my first match point, I played really well," said Rublev.
"I did everything amazing and he just played an unreal rally. He made the backhand down the line and I played a good volley. Then he makes a forehand pass down the line. In most of the matches, normally after these points, they win - it's a turning (point).
"I then started to get a bit more tight and more emotional, I was a bit unlucky. Then when I lost my serve, I said 'Okay, whatever, if I'm gonna lose, at least just try to do your best until the end'."
This was his first win against a Top 10 player since August last year.
Rublev's reward for his fourth semi-final appearance in Qatar is a meeting with Felix Auger-Aliassime.
The Canadian was leading his quarter-final against Daniil Medvedev 6-3 when the Russian had to retire.