Murray lost the first set but came from behind in a three-set game early on Thursday morning, and he will now Tomas Berdych in the last four of the championship.
The victory for the 27-year-old Scot over world number 36, Thiem, made it Murray's 500th career victory, over South Africa's Kevin Anderson, made a sluggish start in his first meeting with 21-year-old Thiem.
"I was getting myself into break-point positions and it is a fine line between going for winners and trying to finish the point and making your opponent play.
"Being a good match player is understanding when to go for your shots and when to back off a little bit."
Andy Murray's opponent in the semi finals, Berdych beat Argentine Juan Monaco 6-3 6-4 in his quarter-final.
Meanwhile, World number one, Serena Williams earned her 700th victory of her career with a victory over Germany's Sabine Lisicki to progress to the semi's of the tourney.
The 17 grand slam title holder defeated 27th seed Lisicki 7-6 (7-4) 1-6 6-3 and reach the last four.
The American became only the eighth player to win 700 WTA matches, although she is still well behind Martina Navratilova with 1,442.
She faces Romania's Simona Halep for a place in the final after the third seed beat American Sloane Stephens 6-1 7-5.
Williams, 33, was set to play Halep in the semi-finals at Indian Wells earlier this month but pulled out with a knee injury, before Halep beat Jelena Jankovic in the final.
Victory for Williams, who is seeking an eighth Miami title, extended her winning run to 19 matches this season.
Lisicki, 25, failed to convert a set point in the first set before losing the tie-break, but recovered from being broken in the first game of the second set to win the next six games.
However, Williams recovered to seal victory in just over two hours.
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