Rublev came through 6-4, 1-6, 6-2 against 19-year-old Frenchman Arthur Fils in a rollercoaster one hour and 52 minutes of highly entertaining tennis at Victoria Park.
The Russian top seed will face another teenager in the last four after "Jerry" Shang Juncheng breezed past world number 16 Tiafoe 6-4, 6-4 to notch the biggest win of his fledgling career and reach his first ATP Tour semi-final.
Rublev raced away with the first set before rising star Fils turned the match on its head with some exceptional attacking play in the second.
Rublev found his range again at the start of the decider to take a 3-0 lead as the youngster from Paris got into a running row with the umpire after some failed challenges to tight calls.
The normally fiery Rublev was, by contrast, calmness personified at the other end and sealed the match with a second break of serve to keep his warm-up for the Australian Open, beginning on January 14th, on track.
"I was just trying to be positive to keep saying to myself, I'm doing really well," Rublev said of the second set onslaught from Fils.
"And if he keeps playing the same way, only congrats to him.
"There is not much you can do when he's serving like that and shooting full power from both sides."
Tiafoe was always under the pump against tireless Shang who broke in the fifth game and was always in control.
Shang had reached the quarter-final by prevailing in two three-set marathons to oust higher-rated opponents, the seventh seed and 33rd-ranked Laslo Djere and world number 50 Botic van de Zandschulp.
The world number 183, who on Thursday was awarded a wildcard into the Australian Open main draw, showed no signs of being overawed, serving out to take the opener without facing a single break point.
Shang was oozing the fearlessness of youth as he edged 3-2 ahead in the second when Tiafoe dropped his serve again.
The left-hander grew in stature the deeper the match went and eased to a famous victory, the best of his fledgling career, on his first match point after 89 minutes.
"Really tough match against Frances," said Shang.
"I played him twice before and went down both times.
"Today was really a mindset change for me. Sometimes you gotta respect the player a little bit less. He's an amazing player and amazing person."
Saturday's other semi-final will be an all-unseeded affair that pits Finland's Emil Ruusuvuori against Austria's Sebastian Ofner.
Ruusuvuori, the world number 69, looked comfortable throughout as he saw off Russia's 67th-ranked Pavel Kotov 6-3, 7-6(4) in one hour and 48 minutes.
Ofner, the world number 43 from Austria, beat Spanish veteran and eighth seed Roberto Bautista Agut 6-4, 6-4 win in 84 minutes.
