The American, ranked No.30, fought past the world No.5 in a 7-5 6-7(3) 10-5 victory that leaves Europe 11-7 up on World.
With three points for a win on the final day, John Isner could secure overall victory for World in the following singles match – although he must overcome 20-time Grand Slam champion, and local hero, Roger Federer.
“We’ve been battling ever since this Laver Cup started,” said Team World Captain John McEnroe.
“Expect the unexpected – here we are now, one more and we got this thing.”
This has to be one of the biggest wins of my career. It means so much more when you’re playing for other people as well – Taylor Fritz
Isner has already enjoyed success on the final day, teaming up with Jack Sock to defeat Federer and Stefanos Tsitsipas in the opening doubles.
Two players who had not expected to be involved on Day 3 were then thrust into the singles spotlight when first Rafael Nadal withdrew because of a hand injury, and then Nick Kyrgios pulled out with a shoulder problem.
The injured duo remained central figures among the courtside supporters but it was left to Thiem and Fritz to battle it out, and for the seventh time in 10 matches at Palexpo, a match tiebreaker was required.
Fritz began the brighter, getting an early break with a deft drop shot to take the first set and pushing hard early in the second.
It took a huge effort from Thiem to hold off the American from 0-40 down in the second set, and by the time the tiebreaker came around the Austrian finally clicked into gear.
Rasping winners and a pumped-up blue bench suggested Thiem might ride the wave all the way through the following match tiebreaker, but it was credit to Fritz that he managed to wrestle back control.
The 21-year-old American’s huge forehand began to dominate, helping him stretch away to 6-2 and again proving too much for Thiem on match point.
“That was such a big match for Team World to win, we really needed that,” said Fritz.
“Now we’re four points in the lead, one match from the win. It was so crucial.
“This has to be one of the biggest wins of my career. It means so much more when you’re playing for other people as well.”
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