The 24-year-old Greek, chasing a maiden Grand Slam title, will next play two-time Wimbledon champion Andy Murray on Centre Court on Thursday.
They both refused to buckle in the fourth, taking it into a tiebreak where Thiem prevailed to force a decider.
Thiem had one chance to break at 3-3 but then handed a match point to Tsitsipas a little later with a double fault at 6-5.
He saved another at 9-7 in the tiebreak before the Greek sealed it on the third attempt.
Reporting by Karolos Grohmann; Additional reporting by Martyn Herman Editing by Toby DavisOur Standards: The Thomson Reuters Trust Principles.
Persons:
Tsitsipas, Austria's Dominic Thiem, Andrew Couldridge, Dominic Thiem of, Andy Murray, Dominic, I'm, I've, Thiem, Roger Federer, Rafa Nadal, Novak Djokovic, Rain, Karolos Grohmann, Martyn Herman, Toby Davis
Organizations:
Lawn Tennis, Croquet Club, Austria's Dominic Thiem REUTERS, Wimbledon, Centre, Thomson
Locations:
London, Britain, Dominic Thiem of Austria, Murray, Madrid