The Briton was broken three times in losing the first set in 34 minutes.
But although he battled back to level, Austrian world number nine Thiem won in two hours and 13 minutes. He will now face Rafael Nadal or Horacio Zeballos.
Elsewhere, British number four Aljaz Bedene beat Serb Laslo Djere 6-2 6-4 to reach the Hungarian Open final.
Murray feels the strain
Having lost both previous meetings with Murray on hard courts, 23-year-old Thiem found the Spanish clay much more to his liking as he secured his first victory against a world number one.
It appeared his array of sparkling winners had been ended by Murray’s determined rearguard, which saw the three-time Grand Slam winner twice stave off break points to take the second set in 48 minutes.
However, in a fluctuating deciding set, Murray, whose three-set victory over Albert Ramos-Vinolas on Friday took a minute less than three hours, hesitated over an overhead smash and instead sent a wild forehand way beyond the baseline, which took Thiem to match point.
Thiem’s impressive victory was duly completed when the Scotsman’s lob drifted wide.
Bedene marches on
There was better news for Bedene, who broke twice in the opening three games of his match against world number 184 Djere.
Having wrapped up the set in 34 minutes he then broke for a 4-3 lead in the second and completed victory in an hour and 19 minutes.
The Slovenia-born 27-year-old, world ranked 68, captured back-to-back titles on the second-tier ATP Challenger Tour earlier this month and has now won 23 of his last 24 matches.
Britain’s Dom Inglot, partnering Robin Haase of the Netherlands, was beaten in three sets by third seeds Juan Sebastian Cabal and Robert Farah of Colombia in the Hungarian Open doubles semi-finals.
