Luke Littler, the world number one, secured his place in the last eight of the tournament held at City of Derry Airport after overcoming Nathan Aspinall in a closely contested match.
The contest finished with a scoreline of 10-9 following a dramatic conclusion in the final leg.
Littler needed to break Aspinall's throw to win the decisive frame, launching from his seventh 180 of the encounter and checking out 58 for a match-clinching 12-darter.
Trailing by one point at 9-8 in a best-of-19 format that remained tight throughout, Littler held the throw in the 18th leg to force a decider.
Clayton beat Japan's Motomu Sakai 10-8 and is one of two Welshmen in the last eight, with Gerwyn Price also progressing courtesy of a 10-5 win against Canada's Jim Long.
England's James Wade was the other winner in Saturday's evening session, defeating Wessel Nijman of the Netherlands, and will face Price next.
The event continues to draw significant attention as competitors from across the globe vie for a place in the championship bracket.