The top seed is aiming to win his home tournament for the first time and, although his match with Andujar lasted close to two hours, Thiem came out on top 7-6 (7-4) 6-4 on Thursday.
Thiem, a beaten finalist at Roland Garros earlier this year but a winner in Barcelona and at Indian Wells, excelled on the second serve and broke four times to overcome the Spaniard.
Lorenzo Sonego, who beat third seed Fernando Verdasco 6-4 6-4, awaits Thiem in the final four.
Joining Thiem and Sonego is 20-year-old Norwegian Casper Ruud, who continued his fine run with an impressive 6-3 7-6 (7-4) victory over Pablo Cuevas to reach his third ATP Tour semi-final of 2019.
Jeremy Chardy beat second seed Dusan Lajovic in the round of 16, but the Frenchman could not follow that win up against Albert Ramos-Vinolas, who triumphed 6-3 6-4 to set up a semi against Ruud.
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