Davis, ranked 20th, made a 138 break in the final frame - the highest of the tournament so far - as he triumphed 6-4 over the twice world champion.
Davis seemed to touch a red with his cue before that ultimate clearance while playing away from the pack of reds, but was not penalised.
However it would likely have made little difference to the outcome. He led 3-1 at the interval and could have gone in 4-0 ahead had he taken all his chances.
When the players returned, Williams had a chance to get back to 3-2 but Davis cleared up with a 50 to snatch the frame before 'Welsh Potting Machine' Williams had a run in the seventies to make it 4-2.
Davis then moved three frames ahead before Williams closed again to 5-3. Davis missed a chance to win the match in frame nine, but dominated the one following to earn a spot against Marco Fu, who beat Martin Gould 6-5 in the Tuesday afternoon session, in the last 16.
Mark Allen beat Robert Milkins 6-2 to set up a meeting with Cao Yupeng, who shocked John Higgins on Monday.
Stephen Maguire won 6-3 against Jamie Burnett and will play Peter Ebdon in the second round. Ebdon received a walkover after being drawn against the suspended Stephen Lee.
In the evening session Ding Junhui won 6-1 against Zhou Yuelong, Judd Trump moved past Fergal O'Brien 6-3 and Ricky Walden set up a match against Mark Selby with a 6-4 result against Lu Ning.
Stuart Bingham, who was in sensational form in qualifying for the semi-finals of the Premier League recently, was shocked 6-4 by Indian world number 72 Aditya Mehta.
The International Championship is the richest of the five ranking events held in China, with prize money of £600,000.
Mark Williams 4-6 Mark Davis
Mark Allen 6-2 Robert Milkins
Stephen Maguire 6-3 Jamie Burnett
Martin Gould 5-6 Marco Fu
Ding Junhui 6-1 Zhou Yuelong
Judd Trump 6-3 Fergal O'Brien
Ricky Walden 6-4 Lu Ning
Stuart Bingham 4-6 Aditya Mehta