Murray, the top seed, lost the opening set in convincing fashion to the Austrian, but recovered to level the match after the second.
The two players exchanged breaks of serve in the deciding set but the fourth seed managed to do enough to seal his place in the final.
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Thiem will face either third seed Rafael Nadal or Horacio Zeballos of Argentina in Sunday's showpiece.
Arguably the biggest win of his career so far, it was Thiem's first triumph over Murray.
The 23-year-old, having reached the final in Barcelona for the first time, will have a shot at a ninth career title on the ATP Tour.
Thiem has become the first Austrian to reach the final of the tournament since Thomas Muster in 1996.
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