“I played all right there,” Clayton told Sky Sports. “Total respect to Stephen, he didn’t play his best game but I kept him under pressure at times. I played well.”
Earlier in the evening, Wade found himself 4-1 down to Van Veen, despite dominating almost every leg as poor finishing cost him.
But the Englishman came out with renewed determination after the break and won the next eight legs on the spin to take control.
Although Van Veen closed the gap and finished the match with the better average – 99.24 to 96.35 – and a better checkout percentage, Wade did enough to see it through.
“I thought I’d thrown it away,” the 42-year-old told Sky Sports. “Towards the end, he was coming back and it was God’s gift to me, not my ability [that secured the win].
“I had a very stern word with myself [after the first session], I was disgusted. You do what you need to do.
“I’m lucky he didn’t make the most of his opportunities. But then if I’d made the most of mine, I’d have been 5-0 or 10-0 up. I’m proper buzzing.”
The third and fourth quarter-finals take place at Winter Gardens on Friday evening with world champion Luke Littler facing Andrew Gilding, while Wales’ Gerwyn Price takes on Josh Rock of Northern Ireland.