The Belgian number one - starring in his third quarter-final - fought back from 3-2 down, winning three consecutive sets to break new ground in the capital.
Clayton's clinical combination finishing proved decisive in the opening exchanges, but Van den Bergh's superior scoring power catapulted him to victory, as he landed 13 maximums to triumph.
"I'm one step closer to make my dream come true of becoming a PDC World Champion," said Van den Bergh, who becomes the first Belgian to reach the semi-finals of the sport's flagship event.
"It was a battle up there today, I had a few moments when I felt comfortable but I missed my chances in the early stages.
"Jonny had chances too but thankfully he didn't take them and I'm delighted to be through.
"It is special to finally pull through to the semi-finals but I still haven't shown my best yet; I think I have the game to go all the way."
Meanwhile, Gabriel Clemens continued his history-making run at the World Darts Championship with a sensational 5-1 demolition of world number one Gerwyn Price.
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 Van den Bergh after their wins over Dobey and Jonny Clayton respectively.
2022/23 World Championships 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