"The level was really high tonight," Wawrinka, 31, said in an on-court interview.
"I'm really happy the way I was playing, the way I found a way to win it."
Thiem kept pace with the Swiss on the strength of his booming forehand, but by the end of the two hour 31 minute match, he lost the range on his favoured shot, belting three forehands well long to lose his way in the third-set decider.
"The player who was the most aggressive, was dictating, was winning most of the points," said Wawrinka, who saved eight of 11 break points. "I'm really happy to have served well in the tiebreak. I think that was the key."
Wawrinka broke in the second game to storm out to a 3-0 lead. Thiem then swept the next three games to level the set on the way to the decider.
The Swiss advanced to his first final four at Indian Wells, where he will meet 21st seed Pablo Carreno Busta, who was also extended to a third-set decider in his 6-1 3-6 7-6(4) win over Pablo Cuevas of Uruguay.
Carreno Busta saved two match points before booking a berth in his first ATP World Tour Masters 1000 semi-final.
The Spaniard ended Cuevas' seven-match winning streak with victory in two hours and five minutes, avenging his defeat to the Uruguayan in Sao Paulo two weeks ago.
"This tournament is one of the best tournaments in the year and maybe one of the toughest, because all the people are playing here," said Carreno Busta.
Wawrinka notched the 450th win of his career in beating Thiem and is bidding to win his second ATP World Tour Masters 1000 crown to add to the 2014 Monte-Carlo trophy he raised after defeating Roger Federer.
(Reporting by Larry Fine in New York; Editing by Nick Mulvenney)
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