Clayton had to work hard for his victory, having gone 2-1 down before having his throw broken in the fourth set.
But the Welshman recovered to clinch a last-leg decider, Harrysson missed chances in the fifth set to take a 3-2 lead, while five of the six sets played went to a deciding leg.
He then checked out on double 16 in the final leg of set six to secure the victory.
Clayton’s victory means he is provisionally up to fourth in the PDC world rankings and would qualify automatically for next year’s Premier League if he is still in that spot at the end of the tournament.
Speaking after his win, Clayton said:
“There are a lot of things on my mind, obviously being world number four was one of them.
“To be fair to Andreas, every time I looked up, he was hitting doubles.
“That was tough. Okay, I didn’t play my best, a lot of loose darts, but that’s what pressure does to you. He kept me under pressure all the time.
“We all go up there, we all miss doubles, we all miss big numbers, it’s cruel at times. But, thank God, I’ve done alright today.”
The fourth round will be completed later on tonight, with three more quarter-finalists to be decided.
Clayton will face Searle in the quarter-final of the competition, with a clash against reigning champion Luke Littler or Kryzsztof Ratajski on the cards in the last four for the winner.
