"I missed a few blacks off the spot but otherwise I thought I played OK," said Williams, who won the championship in 1997.
"I look forward to playing Steve again. We've had a bit of banter in the last few weeks and I called him a `back door Charlie' for getting a wild card into the Champions Cup.
"But Steve beat me in Croydon. And then he gave me the same nickname after I scraped into the semi-finals only because Steve lost to Stephen Hendry in the last group game."
Davis's Champions Cup victory was his first in five attempts against the world No 3 though twice last season their matches were determined only by a deciding frame.
"I've always preferred playing the quicker players, like Mark," said Davis. "Even in the old days I looked forward to meeting the likes of Jimmy White and Alex Higgins more than playing Cliff Thorburn and Eddie Charlton. There is less concentration required."
Williams took a 2-0 lead over Wilkinson with runs of 61, 50 and 80, but the former World Matchplay champion levelled at 2-2 with his own contributions of 86 and 82.
After the interval though Wilkinson scored only 38 points as Williams, as expected, took his allotted place in the draw.
Ken Doherty joined Williams in the fifth round despite a dreadful start against Scotland's Drew Henry.
The 1997 world champion lost the first three frames, gratefully snatched the last one before the break on the black and added the next four frames for a 5-3 victory.
"The schedule wasn't in my favour, having to play last night and first thing today," he said.
Doherty now meets Alan McManus, for a place in the quarter-finals.Reuse content