Thiem, who had won all three of his matches in Kitzbuhel, Austria ahead of his clash with Agut, looked to be cruising through to the final after racing into a 4-1 lead in the first set, having broken the Agut serve twice.
However, Agut showed great resilience to fight back to take a one-set lead following a tie break and silence the socially-distanced crowd - that was the first set Thiem had lost all week at the tournament.
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Thiem was not be outdone, though, with the world number three showing all his class to storm back to take the second 6-2.
There was only going to be one winner from there on in, with Thiem breaking Agut twice early in the final set to set up on one match point, on the Agut serve, and one chance was all Thiem needed, with a blistering return too powerful for his opponent, clinching the final set 6-2 to book a meeting with either Matteo Berrettini or Andrey Rublev in the final.
Weather claimed the second semi-final, which will now take place on Saturday at midday local time.
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