It was an unforgettable clash between Clemens and top seed Price which created a global talking-point after the Welshman sported noise-cancelling ear defenders during the fifth set.
Price had made a blistering start to the contest, taking out 140 and 110 checkouts in consecutive legs to draw first blood with a 107 average in winning the opening set without Clemens having a dart at a double.
However, the German number one responded magnificently, posting ton-plus averages to win each of the next two sets, and then converting a stunning 136 finish en route to a 3-1 lead.
Clemens then fired in a superb 11-darter to wrap up set five, before completing a clean sweep in set six to break new ground at Alexandra Palace.
Cheered on by a significant German contingent at Alexandra Palace, Clemens rose to the occasion, averaging 99.94 and landing eight 180s to take full advantage of a below-par display from Price.
Meanwhile, two-time runner-up Michael Smith will face Clemens in Monday's semi-finals after seeing off Stephen Bunting, while Michael van Gerwen will take on Dimitri Van den Bergh after their wins over Chris Dobey and Jonny Clayton respectively.
"I think this will be amazing for darts in Germany," said Clemens, who continues to pull in record viewing figures on German TV station Sport1, in addition to the coverage through DAZN in his home country.
"This was the biggest match of my life and it's my best performance on the stage. I'm so happy."
He added: "Michael Smith is the Grand Slam champion, one of the favourites for the tournament and one of the best players in the world, but I will have my chances.
"I think it will be a good game. I am only focusing on the semi-finals, and I think I can beat every player."
The semi-finals will take place on a blockbuster Monday night, with German trailblazer Clemens up against fourth seed Smith for a place in Tuesday's showpiece.
Van Gerwen will then continue his bid for a fourth World Championship crown against Van den Bergh, who is aiming to become the first Belgian player to lift the coveted Sid Waddell Trophy.
2022/23 World Darts Championship results and schedule
Sunday January 1 - Quarter-Finals
Dimitri Van den Bergh 5-3 Jonny Clayton (0-3, 3-0, 1-3, 3-0, 2-3, 3-2, 3-1, 3-2)
Michael Smith 5-3 Stephen Bunting (3-2, 0-3, 3-1, 3-1, 3-0, 0-3, 1-3, 3-1)
Gabriel Clemens 5-1 Gerwyn Price (0-3, 3-1, 3-1, 3-1, 3-1, 3-0)
Michael van Gerwen 5-0 Chris Dobey (3-1, 3-1, 3-0, 3-0, 3-1)
Monday January 2 - Semi-finals (1930 GMT)
Gabriel Clemens v Michael Smith
Dimitri Van den Bergh v Michael van Gerwen
Best of 11 sets
Tuesday January 3 - Final (2000 GMT)
Clemens/Smith v Van den Bergh/Van Gerwen
Best of 13 sets