Tsitsipas came out firing to dominate the first set, pouncing on a shallow ball from Rublev to crush a forehand winner for the early break and a 3-1 lead as the fiery Russian's temper began to boil over.
The Greek claimed the set when Rublev failed to put one of his second serves into play and it appeared the Russian might go quietly from there.
Instead, Rublev cut down on the errors in the second set and let out a mighty roar at Pala Alpitour when he broke Tsitsipas for a 5-3 lead.
Rublev saved a break point in the next game and held his fist up high when Tsitsipas' forehand went wide to level the contest at a set apiece.
From there it was Tsitsipas who struggled to keep his composure as Rublev continued to apply pressure, breaking for 2-1 in the third set and again for 5-2 on back-to-back double faults by the world number three.
"I just kept telling myself to keep fighting, keep playing, and we'll see what happens," Rublev said.
"I was happy that I was mentally strong today and I was able to turn the match around because Stef in the beginning was hitting super hard, hitting so many forehand winners."
Next up for Rublev is a semi-final against Norway's Casper Ruud (23).
