The two-time French Open semi-finalist recovered from a set down to book a last-16 clash against either Novak Djokovic or Borna Coric with a 5-7, 7-5, 7-5 victory.
Rublev spurned a match point on his own serve late in the deciding set as he fired a forehand narrowly wide.
"I had to fight from the first to the last point, which of course I did well. But I was also lucky at the end when he had a match point and missed a forehand by 10 centimetres or so," said Thiem.
Austria's Dominic Thiem celebrates after winning his tennis match against Russia's Andrey Rublev during the Monte-Carlo ATP Masters Series Tournament in Monaco on Tuesday — AFP