The 18-year-old came through 6-4, 7-6(6) against the fifth-ranked Rublev, a day after defeating three-time Grand Slam title winner Andy Murray.
"It's just been an incredible week. From the beginning, I played very well and I knew I could play with the big players. It's an amazing feeling to reach the semi-finals after beating those good players," said wildcard Mensik who arrived in the Gulf ranked at 116.
"But the job is not done yet. Hopefully, I can play like this again in the semis and go on to make the final."
With his one-hour, 38-minute win on Thursday, Mensik became the youngest player to defeat a top-five player since Carlos Alcaraz overcame Stefanos Tsitsipas at the 2021 US Open.
Mensik will face Gael Monfils, who beat Ugo Humbert in an all-French quarter-final, for a place in Saturday's final.
The other semi-final will see Australia's Alexei Popyrin face Russian second seed Karen Khachanov.
Popyrin eased past Kazakh fourth seed Alexander Bublik 6-4, 6-4 while Khachanov went through when Finnish opponent Emil Ruusuvuori retired with an injury after just three games.
