The Greek, chasing a maiden Grand Slam title, will next play two-time Wimbledon champion Andy Murray (36).
The match had started on Tuesday on Court Two with Thiem, who in June was getting match practice in the lower-tier Challenger events after an injury, broke Tsitsipas midway through the opener to snatch the first set.
The Austrian's big backhand was again looking more like the weapon it once was rather than the liability it had become since his wrist injury.
Thiem broke the Grand Slam stranglehold of Roger Federer, Rafa Nadal and Novak Djokovic by winning the 2020 US Open but injury sidelined him for months on end and his ranking slipped to number 352 in June last year.
Now ranked 91st, Thiem showed no signs of stage fright as he battled against the fifth seed who had beaten him in Madrid earlier this year.
Rain forced play to be suspended and it resumed more than 24 hours later, on Wednesday afternoon, with the world number six levelling with a confident tie-break win and looking sharper than a day earlier.
He did squander five break points at 1-1 in the third set but earned his first break of the match a little later to take control, clinching the third set.
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 tie-break before the Greek sealed it on the third attempt.
