The Austrian saved a match-point in the tenth game of the deciding set before inflicting misery on his explosive rival – and he will challenge either Novak Djokovic or Borna Coric on Thursday.
It was a rip-roaring affair between two talented shotmakers at ease on the red dirt. Rublev faltered with the winning line in sight, though, and he was eventually denied just a second Top 10 victory.
Thiem, who enjoyed a sterling clay-court campaign last year, was playing his first match since recovering from a foot problem that forced him to miss the Miami Open.
His lack of match-practice in recent weeks was apparent in the early stages, and he dropped serve in the 11th game to concede a competitive opener.
The 24-year-old made more of an impression in his opponent’s service games in the second set, however it still took him until the 12th game to make his superiority count.
The third set proved to be a thrilling finale to an engrossing duel. Rublev fell behind early but responded impressively and levelled at four apiece.
The Russian had wrestled back momentum, and a venomous backhand passing-shot winner lashed down the line earned him the opportunity to serve for the match.
Rublev was closing in on arguably the biggest win of his career, but he clipped a forehand wide to squander his only match-point chance. Thiem capitalised on his rival’s generosity and restored parity with a well-directed forehand passing-shot.
The Russian appeared crestfallen at his inability to apply the finishing touches – and he never recovered. In his next service game, he committed a double-fault to bring an unsatisfactory ending to a pulsating affair.
Elsewhere on the grounds, Gilles Simon started his campaign with a 6-3 4-6 6-2 success over Adrian Mannarino while qualifier Marco Cecchinato upset Damir Dzumhur 6-3 6-2.
In other news, Jared Donaldson has been fined 5,000 euros (£4,300) for unsportsmanlike conduct during his loss to Albert-Ramos Vinolas on Monday.
The American vented his fury towards umpire, Arnaut Gabas, in a confrontational manner after a decision was wrongly ruled against him.